MathsICSE Class 8 · Chapter 3

Squares and Square Roots ICSE Class 8: Factor and Long Division Methods

Q: Which method should I use, prime factor or long division?

Use prime factors when the number breaks easily into small primes (like 1764 or 2025). Use long division when factors are large (like 5329 = 73²), when there is a decimal, or when the question asks for a remainder.

Q: Does a number ending in 1, 4, 5, 6 or 9 have to be a perfect square?

No. Those endings are only possible for squares, not proof of one. For example 14 ends in 4 but is not a perfect square. Use prime factors or long division to be sure.

Q: Why must the zeros at the end of a square be even in number?

Each 10 at the end comes from a factor 10 = 2 × 5. In a square every factor appears in pairs, so the zeros also come in pairs.

Q: Do I need square roots of non-perfect squares in Class 8?

Only in the sense of finding the nearest perfect square (least number to add or subtract). Approximate roots to many decimal places are not part of the Class VIII syllabus limits.

Q: What is the square root of a number with 2 decimal places?

If it is a perfect square, its root has 1 decimal place: √12.25 = 3.5.

Updated 15 October 2026 · Fact-checked

The square of a number n is n × n, and the square root of a number is the value that multiplied by itself gives that number. ICSE Class 8 finds square roots by the prime factor method and by the long division method, for numbers of up to four digits and up to two decimal places.

What Squares and Square Roots is about

When a number is multiplied by itself the result is its square: $7^{2} = 49$ . Reversing this gives the square root: $49 = 7$ . Numbers like 1, 4, 9, 16, 25, ... that are squares of whole numbers are called perfect squares.

The CISCE Class VIII curriculum asks you to find squares and square roots by two methods:

the prime factor method (pair up equal prime factors), and
the division method (also called the long division method),

for numbers containing no more than 4 digits in total and no more than 2 decimal places, such as 7056, 56.25 or 13.69.

In a school exam you will test whether a number is a perfect square, find the smallest number to multiply or divide by to make a perfect square, find the least number to add or subtract, take square roots of decimals and fractions, and solve problems about square fields, tiles and rows.

Weightage: CISCE publishes no question paper, marking scheme or chapter weightage for Class VIII Mathematics: your school sets and marks the exam. Squares and Square Roots sits in the Number System theme of the CISCE Curriculum for Upper Primary Classes (VI-VIII).Source: CISCE Curriculum for Upper Primary Classes (VI-VIII), RDCD 2016

Key concepts

Perfect squares and their properties

A perfect square is the square of a whole number. Useful facts that let you reject a number quickly:

A perfect square ends in 0, 1, 4, 5, 6 or 9. It never ends in 2, 3, 7 or 8.
If it ends in zeros, the number of zeros is even: $40 0^{2} = 160000$ has 4 zeros.
The square of an even number is even; the square of an odd number is odd.
In the prime factorisation of a perfect square, every prime appears an even number of times.

The last test is the decisive one. The first three only tell you when a number is definitely not a perfect square.

Worked example

Is 2352 a perfect square? If not, find the smallest number by which it must be multiplied to get a perfect square, and find that square root.

Try it first, then show the solutionSolution

$2352 = 2^{4} \times 3 \times 7^{2}$ .
The prime 3 appears only once, so 2352 is not a perfect square.
Multiply by 3: $7056 = 2^{4} \times 3^{2} \times 7^{2}$ .
$7056 = 2^{2} \times 3 \times 7 = 84$ .

Answer: No; multiply by 3; $7056 = 84$ .

Square root by the prime factor method

Steps:
1. Write the number as a product of primes (divide by 2, 3, 5, 7, ... repeatedly).
2. Group the primes into pairs of equal factors.
3. Take one prime from each pair and multiply them.

The product is the square root. If a prime is left without a partner, the number is not a perfect square.

This method is quick when the prime factors are small.

Worked example

Find $1764$ by the prime factor method.

Try it first, then show the solutionSolution

$1764 = 2 \times 2 \times 3 \times 3 \times 7 \times 7$ .
Pairs: $(2 \times 2) (3 \times 3) (7 \times 7)$ .
One from each pair: $2 \times 3 \times 7 = 42$ .

Answer: $42$

Making a number a perfect square

After prime factorisation, look for the primes that have no partner.

Smallest multiplier: multiply by the unpaired primes. This gives each of them a partner.
Smallest divisor: divide by the unpaired primes. This removes them.

Then find the square root of the new number from the pairs that remain.

Worked example

Find the smallest number by which 1008 must be divided to get a perfect square. Find the square root of the result.

Try it first, then show the solutionSolution

$1008 = 2^{4} \times 3^{2} \times 7$ .
7 has no partner, so divide by 7: $1008 \div 7 = 144$ .
$144 = 2^{4} \times 3^{2}$ , so $144 = 2^{2} \times 3 = 12$ .

Answer: Divide by 7; the square root is 12.

Square root by the long division method

Use this when factors are large or awkward.
1. Put bars over pairs of digits from the right (for 5329: $\overline{53} \overline{29}$ ).
2. Find the largest number whose square is at most the first group. Write it as the first digit of the root and subtract its square.
3. Bring down the next pair. Double the root found so far and write it with a blank: $14_$ .
4. Find the largest digit $d$ so that $14 d \times d$ is at most the new dividend. Write $d$ in the root, subtract, and repeat.

If the final remainder is 0, the number is a perfect square.

Worked example

Find $5329$ by the long division method.

Try it first, then show the solutionSolution

Pairs: $\overline{53} \overline{29}$ . $7^{2} = 49 \leq 53$ ; remainder $53 - 49 = 4$ .
Bring down 29: dividend 429. Double 7 to get 14.
$143 \times 3 = 429$ , remainder 0.
Root $= 73$ .

Answer: $73$

Square roots of decimals and fractions

Decimals: pair the digits of the whole-number part from the right of the decimal point and the decimal part from the left of it. Put the decimal point in the root as soon as you cross the point in the number. A number with 2 decimal places has a square root with 1 decimal place: $17.64 = 4.2$ because $4. 2^{2} = 17.64$ .

Fractions: $\frac{a}{b} = \frac{a}{b}$ . Change a mixed number to an improper fraction first: $2 \frac{14}{25} = \frac{64}{25} = \frac{8}{5}$ .

Worked example

Find $17.64$ .

Try it first, then show the solutionSolution

Pairs: $\overline{17} . \overline{64}$ .
$4^{2} = 16 \leq 17$ ; remainder 1. Bring down 64 (crossing the point, so put the decimal point in the root): 164.
Double 4 = 8: $82 \times 2 = 164$ , remainder 0.
Root $= 4.2$ .

Answer: $4.2$

Least number to add or subtract

Do the long division for the given number.

If there is a remainder, that remainder is the least number to subtract to get a perfect square (the square of the root you found).
The least number to add is $(root + 1)^{2}$ minus the given number.

You can check either answer by squaring.

Worked example

Find the least number that must be subtracted from 1300, and the least number that must be added to it, to get a perfect square.

Try it first, then show the solutionSolution

Long division gives root 36 with remainder $1300 - 3 6^{2} = 1300 - 1296 = 4$ .
Subtract 4 to get $1296 = 3 6^{2}$ .
Next square: $3 7^{2} = 1369$ ; add $1369 - 1300 = 69$ .

Answer: Subtract 4, or add 69.

Maths in real life: Squares and Square Roots

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Side from area

Suppose a square carpet covers exactly 6.25 m² of floor.

How long is each side of the carpet?

Try it first, then show the solutionSolution

Side $= 6.25$ .
$2. 5^{2} = 6.25$ , so the side is 2.5 m.

Answer: 2.5 m

Memory hook: Area of a square is side squared, so side is the square root of the area.

Rows equal to columns

Imagine 529 chairs are to be arranged for a school assembly with as many rows as there are chairs in each row.

How many rows are there?

Try it first, then show the solutionSolution

Rows × chairs per row $= 529$ with both equal, so rows $= 529$ .
$529 = 23 \times 23$ , so there are 23 rows.

Answer: 23 rows of 23 chairs

Memory hook: A square arrangement is a perfect square.

Nearest perfect square

Suppose you have 120 square tiles and want to lay the largest possible square, or buy just enough to make the next size up.

What square can you lay now, and how many tiles must you buy for the next size?

Try it first, then show the solutionSolution

$1 0^{2} = 100 \leq 120 < 121 = 1 1^{2}$ .
You can lay a $10 \times 10$ square (20 tiles spare), or buy $121 - 120 = 1$ tile for an $11 \times 11$ square.

Answer: A 10 × 10 square now; 1 more tile makes 11 × 11.

Memory hook: Squeeze the number between two neighbouring squares.

Topics in this chapter

Squares and Square Roots formula sheet

Square: $n^{2} = n \times n$; A perfect square has only even powers in its prime factorisation.
Square root: $n^{2} = n$; For n ≥ 0.
Root of a product: $ab = a \times b$; Basis of the prime factor method.
Root of a fraction: $\frac{a}{b} = \frac{a}{b}$; Use improper fractions.
Least number to add: $add = (r + 1)^{2} - N$; r is the whole-number root found by long division.

How the school exam asks Squares and Square Roots

The question patterns to expect, and the method that works for each.

Question type	How it is asked	Method
Perfect square or not	Which of these numbers are not perfect squares? Give a reason.	Check the last digit and trailing zeros; confirm with prime factors.
Root by prime factors	Find the square root of 2025 by the prime factor method.	Factorise, pair, take one from each pair.
Root by long division	Find √7056 or √56.25 by the division method.	Pair digits from the decimal point; double, guess the digit, subtract.
Smallest multiplier or divisor	Find the smallest number by which 2028 must be multiplied to give a perfect square.	Unpaired primes are the answer.
Word problems	Square fields, tiles, rows of plants, collections where each person gives as much as the number of people.	Spot the square: side × side, rows × plants per row, people × amount each.

Tricks to remember Squares and Square Roots

Short memory aids. Say each one aloud once, then use it on a question the same day.

2, 3, 7, 8 never end a square
Squares end in 0, 1, 4, 5, 6, 9. Think '2378 is never great'.
Use it when: Rejecting a number in one look.
Pair and pick one
In the prime factor method, every pair sends one member to the answer.
Use it when: Finding the root from prime factors.
Double, guess, multiply, subtract
Long division has the same four moves every round: Double the root, Guess the digit, Multiply, Subtract.
Use it when: Not losing your place in long division.
Half the decimal places
The root has half as many decimal places as the perfect square: 2 places become 1.
Use it when: Placing the decimal point.

Squares and Square Roots: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Example 1: Spot the non-square · 1 mark · MCQExam level

Which of these is not a perfect square?
(a) 1024
(b) 1057
(c) 1089
(d) 1296

Try it first, then show the solutionSolution

A perfect square never ends in 7.
Check the rest: $3 2^{2} = 1024$ , $3 3^{2} = 1089$ , $3 6^{2} = 1296$ .

Answer: (b)

Example 2: Prime factor method · 3 marks · Short answerExam level

Find $2025$ by the prime factor method.

Try it first, then show the solutionSolution

$2025 = 3 \times 3 \times 3 \times 3 \times 5 \times 5$ .
Pairs: $(3 \times 3) (3 \times 3) (5 \times 5)$ .
Root $= 3 \times 3 \times 5 = 45$ .

Answer: $45$

Example 3: Long division method · 3 marks · Short answerExam level

Find $7056$ by the long division method.

Try it first, then show the solutionSolution

Pairs: $\overline{70} \overline{56}$ . $8^{2} = 64$ ; remainder 6.
Bring down 56: dividend 656. Double 8 = 16.
$164 \times 4 = 656$ ; remainder 0. Root $= 84$ .

Answer: $84$

Example 4: Smallest multiplier · 3 marks · Short answerHarder exam level

Find the smallest number by which 2028 must be multiplied so that the product is a perfect square. Find the square root of the product.

Try it first, then show the solutionSolution

$2028 = 2^{2} \times 3 \times 1 3^{2}$ .
3 is unpaired, so multiply by 3: $6084 = 2^{2} \times 3^{2} \times 1 3^{2}$ .
$6084 = 2 \times 3 \times 13 = 78$ .

Answer: Multiply by 3; $6084 = 78$ .

Example 5: Decimal by long division · 3 marks · Short answerHarder exam level

Find $56.25$ by the long division method.

Try it first, then show the solutionSolution

Pairs: $\overline{56} . \overline{25}$ . $7^{2} = 49$ ; remainder 7.
Bring down 25 (crossing the decimal point, so put the point in the root): 725. Double 7 = 14.
$145 \times 5 = 725$ ; remainder 0. Root $= 7.5$ .

Answer: $7.5$

Example 6: Least number to subtract · 4 marks · Long answerHarder exam level

Find the least number which must be subtracted from 6203 to obtain a perfect square. Also find the square root of that perfect square.

Try it first, then show the solutionSolution

Pairs: $\overline{62} \overline{03}$ . $7^{2} = 49$ ; remainder 13.
Bring down 03: dividend 1303. Double 7 = 14.
$148 \times 8 = 1184 \leq 1303$ but $149 \times 9 = 1341$ is too big; remainder $1303 - 1184 = 119$ .
Subtract 119: $6203 - 119 = 6084 = 7 8^{2}$ .

Answer: Subtract 119; the square root is 78.

Example 7: The orchard · 5 marks · Long answerHOTS / competency

A gardener has 1000 saplings. He wants to plant them in a square pattern with the same number of rows as plants in each row. (i) What is the largest square he can plant, and how many saplings are left over? (ii) How many more saplings must he buy to plant the next larger square, and how many rows will it have?

Try it first, then show the solutionSolution

Long division of 1000 gives root 31 with remainder $1000 - 961 = 39$ .
(i) Largest square: 31 rows of 31, using 961 saplings; 39 are left over.
(ii) Next square: $3 2^{2} = 1024$ .
Extra saplings needed $= 1024 - 1000 = 24$ .
The new square has 32 rows of 32 plants.

Answer: (i) 31 × 31, 39 left; (ii) buy 24, giving 32 rows.

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

Tiling a square hall

Suppose the floor of a square hall has an area of 33.64 m². It is to be covered with square tiles of side 20 cm, and a wooden skirting strip is to be fixed along the walls except across a door 1 m wide.

(i) · 3 marks
Find the side of the hall by the long division method.
Answer
Answer: 5.8 m
Pairs $\overline{33} . \overline{64}$ : $5^{2} = 25$ , remainder 8; bring down 64 to get 864; double 5 = 10; $108 \times 8 = 864$ . Side $= 5.8$ m.
(ii) · 3 marks
How many tiles fit along one side, and how many tiles are needed in all?
Answer
Answer: 29 along a side; 841 tiles
$5.8$ m $= 580$ cm; $580 \div 20 = 29$ . Total $= 2 9^{2} = 841$ tiles.
(iii) · 4 marks
Skirting costs ₹25 per metre. Find the cost of the skirting.
Answer
Answer: ₹555
Perimeter $= 4 \times 5.8 = 23.2$ m. Less the door: $23.2 - 1 = 22.2$ m. Cost $= 22.2 \times 25 = ₹555$ .

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Find $5776$ by the prime factor method. (3 marks)

Step	Marks
$5776 = 2 \times 2 \times 2 \times 2 \times 19 \times 19$	1
Pairs $(2 \times 2) (2 \times 2) (19 \times 19)$	1
Root $= 2 \times 2 \times 19 = 76$	1

Find $30.25$ by the long division method. (4 marks)

Step	Marks
Correct pairing $\overline{30} . \overline{25}$	1
$5^{2} = 25$ ; remainder 5	1
Bring down 25 to get 525; decimal point placed in the root; double 5 = 10	1
$105 \times 5 = 525$ , remainder 0; root $= 5.5$	1

Find the smallest number by which 1323 must be multiplied to make it a perfect square. Find the square root of the new number. (5 marks)

Step	Marks
Prime factorisation: $1323 = 3 \times 3 \times 3 \times 7 \times 7$	1
Pairs $(3 \times 3) (7 \times 7)$ with one 3 unpaired	1
Multiply by 3: $1323 \times 3 = 3969$	1
$3969 = 3^{4} \times 7^{2}$	1
$3969 = 3^{2} \times 7 = 63$	1

Where students lose marks in Squares and Square Roots

The slips examiners see most often, and what to write instead.

Mistake: Pairing digits from the left in long division
Pairs for the whole-number part must start at the decimal point (or the right end).
Wrong working
$\overline{62} \overline{03}$ written as $\overline{6} \overline{20} \overline{3}$
Right working
$\overline{62} \overline{03}$
Do this: Start pairing at the decimal point and move outwards in both directions.
Usually costs: The whole question.
Mistake: Forgetting to double the root
The new trial divisor must be twice the root so far, with a digit attached.
Wrong working
Using $7_$ instead of $14_$ after the first digit 7
Right working
Use $14_$ and try $148 \times 8$
Do this: Write the doubled number on the left before guessing the digit.
Usually costs: 2 to 3 marks.
Mistake: Multiplying by a paired prime
Only an unpaired prime needs a partner.
Wrong working
$2028 = 2^{2} \times 3 \times 1 3^{2}$ , so multiply by 13
Right working
Multiply by 3, the only unpaired prime
Do this: Circle each pair; the uncircled primes are the multiplier.
Usually costs: Most of the marks.
Mistake: Misplacing the decimal point
$56.25$ is 7.5, not 75 or 0.75.
Do this: Put the point in the root the moment you bring down the first pair after the point, and check by squaring.
Usually costs: 1 mark.
Mistake: Giving the remainder as the number to add
The remainder is what to subtract; the number to add comes from the next square.
Do this: Number to add $= (r + 1)^{2} - N$ .
Usually costs: 1 to 2 marks.

Practice questions on Squares and Square Roots

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQExam level
The square of 400 ends in how many zeros?
(a) 2
(b) 3
(c) 4
(d) 8
Hint
Double the number of zeros.
Answer
(c): $40 0^{2} = 160000$ .
Q2 · 2 marks · Short answerExam level
Find $1225$ by the prime factor method.
Hint
1225 = 5 × 5 × 7 × 7.
Answer
$5 \times 7 = 35$ .
Q3 · 3 marks · Short answerExam level
Find $3249$ by the long division method.
Hint
Start with $5^{2} = 25$ .
Answer
Remainder 7; bring down 49 to get 749; $107 \times 7 = 749$ . Root $= 57$ .
Q4 · 2 marks · Short answerHarder exam level
Find $2 \frac{14}{25}$ .
Hint
Make it an improper fraction first.
Answer
$\frac{64}{25} = \frac{8}{5} = 1 \frac{3}{5}$ .
Q5 · 1 mark · Assertion-reasonHarder exam level
Assertion (A): $1.21 = 1.1$ .
Reason (R): The square root of a perfect square with 2 decimal places has 1 decimal place.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
Square 1.1.
Answer
(c). $1. 1^{2} = 1.21$ , and the root has half the decimal places, as (R) says.
Q6 · 3 marks · Short answerHarder exam level
Find the smallest number by which 1620 must be divided to get a perfect square. Find the square root of the quotient.
Hint
Prime factorise 1620.
Answer
$1620 = 2^{2} \times 3^{4} \times 5$ . Divide by 5: $324 = 1 8^{2}$ .
Q7 · 3 marks · Short answerHarder exam level
Find $13.69$ by the long division method.
Hint
$3^{2} = 9$ ; remainder 4.
Answer
Bring down 69 to get 469; double 3 = 6; $67 \times 7 = 469$ . Root $= 3.7$ .
Q8 · 3 marks · Short answerHarder exam level
Find the least number that must be added to 1750 to make it a perfect square. Find the square root of the new number.
Hint
$4 1^{2} = 1681$ and $4 2^{2} = 1764$ .
Answer
$1764 - 1750 = 14$ ; $1764 = 42$ .
Q9 · 3 marks · Short answerHarder exam level
The area of a square field is 5184 m². Find the cost of fencing it at ₹15 per metre.
Hint
Side = √area.
Answer
Side $= 5184 = 72$ m; perimeter $= 288$ m; cost $= 288 \times 15 = ₹4320$ .
Q10 · 2 marks · Short answerHarder exam level
Find the greatest 4-digit number that is a perfect square.
Hint
Do long division on 9999.
Answer
$9999$ gives 99 remainder 198, so the answer is $9999 - 198 = 9801 = 9 9^{2}$ .
Q11 · 4 marks · Long answerHOTS / competency
Find the smallest 4-digit perfect square that is divisible by each of 6, 9 and 15.
Hint
Find the LCM, then the smallest square multiple of it.
Answer
LCM $= 90 = 2 \times 3^{2} \times 5$ . Smallest square multiple: $2^{2} \times 3^{2} \times 5^{2} = 900$ (3 digits). Next: $900 \times 4 = 3600 = 6 0^{2}$ , which has 4 digits. Answer 3600.
Q12 · 4 marks · Long answerHOTS / competency
Each student in a class gave as many paise as there were students in the class. The total collected was ₹22.09. How many students are in the class?
Hint
Change rupees to paise; total = (students)².
Answer
₹22.09 $= 2209$ paise $= n^{2}$ . Long division: $4^{2} = 16$ , remainder 6; 609; $87 \times 7 = 609$ . $n = 47$ students.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Name the digits a perfect square can never end in.Show answer
2, 3, 7 and 8.
2.What is √1225?Show answer
35.
3.How many zeros does the square of 3000 end with?Show answer
6.
4.What is √0.49?Show answer
0.7.
5.What is √(16/81)?Show answer
4/9.
6.Is 2700 a perfect square?Show answer
No. 2700 = 2² × 3³ × 5², and the prime 3 appears an odd number of times.
7.Find the square of 1.3.Show answer
1.69.
8.Which is the greatest 4-digit perfect square?Show answer
9801 = 99².

Best order to study it

Learn squares from 1² to 30² and the properties of perfect squares.
Practise the prime factor method and the smallest multiplier or divisor.
Learn the long division method on 4-digit whole numbers.
Extend long division to decimals with up to two decimal places, then fractions.
Finish with least number to add or subtract and word problems.

Quick revision checklist

The night before the exam: can you do each of these without looking?

Can you reject a non-square from its last digit?
Can you pair prime factors and pick one from each pair?
Do you pair digits starting from the decimal point?
Do you double the root at every step of long division?
Do you place the decimal point correctly in the root?
Can you find the least number to add and to subtract?
Do you check every root by squaring it?

Squares and Square Roots: frequently asked questions

Which method should I use, prime factor or long division?

Use prime factors when the number breaks easily into small primes (like 1764 or 2025). Use long division when factors are large (like 5329 = 73²), when there is a decimal, or when the question asks for a remainder.

Does a number ending in 1, 4, 5, 6 or 9 have to be a perfect square?

No. Those endings are only possible for squares, not proof of one. For example 14 ends in 4 but is not a perfect square. Use prime factors or long division to be sure.

Why must the zeros at the end of a square be even in number?

Each 10 at the end comes from a factor 10 = 2 × 5. In a square every factor appears in pairs, so the zeros also come in pairs.

Do I need square roots of non-perfect squares in Class 8?

Only in the sense of finding the nearest perfect square (least number to add or subtract). Approximate roots to many decimal places are not part of the Class VIII syllabus limits.

What is the square root of a number with 2 decimal places?

If it is a perfect square, its root has 1 decimal place: √12.25 = 3.5.

Based on: CISCE Curriculum for Upper Primary Classes (VI-VIII), Mathematics (Class VIII); CISCE RDCD: Curriculum Pre-School to Class VIII