MathsICSE Class 8 · Squares and Square Roots

Square Root by Prime Factorisation: Pairing, Multiplier and Divisor (ICSE Class 8)

Updated 17 October 2026 · Fact-checked

Break the number into prime factors and make pairs of equal primes. The square root is the product of one prime from each pair. If a prime is left unpaired, the number is not a perfect square; multiply by the unpaired primes (or divide by them) to make it one.

Understand Square Root by Prime Factorisation

A perfect square is the square of a whole number. The CISCE curriculum asks for square roots by the factor method for numbers of at most four digits in total.

Method. Divide repeatedly by the smallest primes $2, 3, 5, 7, \dots$ until the quotient is $1$ , then write the number as a product of primes. If every prime occurs an even number of times, group them in pairs; the square root takes one prime from each pair.

Example: $2025 = 3 \times 3 \times 3 \times 3 \times 5 \times 5 = 3^{4} \cdot 5^{2}$ , so $2025 = 3^{2} \times 5 = 45$ .

Smallest multiplier. If some primes are left without a partner, multiply the number by each such prime (once) to complete the pairs. The new number is the smallest perfect square that is a multiple of the original.

Smallest divisor. Alternatively, divide the number by the unpaired primes. The result is the largest perfect square that divides it with those unpaired primes removed; this gives the least number to divide by.

Quick tests. A perfect square never ends in $2, 3, 7, 8$ , and the number of trailing zeros is even. These help to rule out candidates, but the prime-factor test is conclusive.

Applications. Side of a square from its area, number of rows when rows equal columns, perimeter and cost of fencing, and smallest perfect square divisible by several numbers (use their LCM and complete its pairs).

Square Root by Prime Factorisation: formulas to remember

Square root from primes: $p_{1}^{2 a} p_{2}^{2 b} = p_{1}^{a} p_{2}^{b}$; Halve every exponent.
Smallest multiplier: $multiply by each unpaired prime once$; Gives the smallest square multiple.
Smallest square multiple of several numbers: $LCM, then complete unpaired primes$; Used for hall, tile and grid questions.

How to solve Square Root by Prime Factorisation questions

Factorise first, pair second, then decide whether to multiply or divide.

1Divide by $2, 3, 5, 7, \dots$ in turn and write the prime factorisation.
2Count each prime; pair up equal primes.
3If nothing is left over, take one from each pair and multiply for the root.
4If primes are left over, multiply (or divide) by exactly those primes.
5Compute the root of the new number and verify by squaring.

Square Root by Prime Factorisation in real life

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.

Square seating

Suppose 2025 chairs are to be placed in a square arrangement with as many rows as chairs per row.

How many chairs go in each row?

Try it first, then show the solutionSolution

$2025 = 3^{4} \times 5^{2}$ .
$2025 = 3^{2} \times 5 = 45$ .

Answer: 45 chairs per row (45 rows).

Memory hook: Pairs of equal primes become one prime in the root.

Tricks to remember Square Root by Prime Factorisation

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

Odd one out
Primes without a partner are the ones you multiply or divide by.
Use it when: Smallest multiplier or divisor questions.
Last digit filter
A square cannot end in 2, 3, 7 or 8.
Use it when: Quick elimination in MCQs.

Square Root by Prime Factorisation: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Direct root · 1 mark · MCQBoard level

$2025$ equals:
(a) 35 (b) 45 (c) 55 (d) 65

Try it first, then show the solutionSolution

$2025 = 3^{4} \times 5^{2}$ , root $= 3^{2} \times 5 = 45$ .
Check: $4 5^{2} = 2025$ .

Answer: (b)

Example 2: Root by factors · 2 marks · Short answerBoard level

Find $1936$ by the prime factor method.

Try it first, then show the solutionSolution

$1936 = 2^{4} \times 1 1^{2}$ (since $1936 = 16 \times 121$ ).
$1936 = 2^{2} \times 11 = 44$ .
Check: $4 4^{2} = 1936$ .

Answer: 44

Example 3: Smallest multiplier · 3 marks · Short answerHarder board level

Find the smallest number by which 1176 must be multiplied to give a perfect square, and the square root of the new number.

Try it first, then show the solutionSolution

$1176 = 2^{3} \times 3 \times 7^{2}$ .
Unpaired: one $2$ and one $3$ . Multiply by $2 \times 3 = 6$ .
New number $= 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2}$ ; root $= 2^{2} \cdot 3 \cdot 7 = 84$ .
Check: $8 4^{2} = 7056$ .

Answer: Multiplier 6; square root 84

Example 4: Smallest divisor · 3 marks · Short answerHarder board level

Find the smallest number by which 3675 must be divided to give a perfect square, and the square root of the quotient.

Try it first, then show the solutionSolution

$3675 = 3 \times 5^{2} \times 7^{2}$ .
Only $3$ is unpaired, so divide by $3$ .
Quotient $= 1225 = 5^{2} \cdot 7^{2}$ , root $= 35$ .

Answer: Divisor 3; square root 35

Example 5: Smallest square multiple of three numbers · 4 marks · Long answerHOTS / competency

Find the smallest perfect square divisible by each of 8, 12 and 15, and its square root.

Try it first, then show the solutionSolution

$8 = 2^{3}$ , $12 = 2^{2} \cdot 3$ , $15 = 3 \cdot 5$ . LCM $= 2^{3} \cdot 3 \cdot 5 = 120$ .
Unpaired primes: $2, 3, 5$ . Multiply $120$ by $30$ : $3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$ .
Square root $= 2^{2} \cdot 3 \cdot 5 = 60$ . Check: $6 0^{2} = 3600$ and $3600 \div 8 = 450$ , $\div 12 = 300$ , $\div 15 = 240$ .

Answer: 3600; square root 60

Where students lose marks in Square Root by Prime Factorisation

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

Mistake: Taking the root of each prime including unpaired ones
Only complete pairs contribute to the root.
Wrong working
$1176 = 2 \times 3 \times 7 \times 2$ .
Right working
$1176 = 2^{3} \cdot 3 \cdot 7^{2}$ is not a perfect square; multiply by $6$ first.
Do this: Count each prime; every count must be even.
Usually costs: All marks if the question asks for a perfect square.
Mistake: Dividing when the question says multiply
The two questions need opposite actions on the unpaired primes.
Wrong working
"Smallest multiplier of 1350": answer given as $1350 \div 6 = 225$ .
Right working
Multiplier is $6$ and the new number is $8100$ (dividing by $6$ gives $225$ , which answers a different question).
Do this: Underline the verb multiplied/divided before starting.
Usually costs: The answer mark.

Exam tips

Show the factor tree or repeated division; examiners credit it.
Verify by squaring the root.
For "smallest square divisible by", take the LCM first.

Practice questions on Square Root by Prime Factorisation

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 · MCQBoard level
Which of these is a perfect square?
(a) 2352 (b) 2304 (c) 2310 (d) 2332
Hint
Check the last digit, then factorise.
Answer
(b). $2304 = 2^{8} \times 3^{2} = 4 8^{2}$ .
Q2 · 2 marks · Short answerBoard level
Find $9604$ by prime factorisation.
Hint
9604 = 4 x 2401.
Answer
$9604 = 2^{2} \times 7^{4}$ , so root $= 2 \times 7^{2} = 98$ . Check: $9 8^{2} = 9604$ .
Q3 · 3 marks · Short answerHarder board level
Find the smallest number by which 1350 should be multiplied to obtain a perfect square, and the root of the product.
Hint
$1350 = 2 \times 675$ .
Answer
$1350 = 2 \cdot 3^{3} \cdot 5^{2}$ ; unpaired: one $2$ and one $3$ ; multiply by $6$ : $8100 = 2^{2} \cdot 3^{4} \cdot 5^{2}$ , root $= 2 \cdot 9 \cdot 5 = 90$ .
Q4 · 3 marks · Short answerHarder board level
By what smallest number should 2880 be divided to get a perfect square? Find the root of the quotient.
Hint
$2880 = 64 \times 45$ .
Answer
$2880 = 2^{6} \cdot 3^{2} \cdot 5$ ; divide by $5$ : $576$ , root $= 2^{3} \cdot 3 = 24$ .
Q5 · 4 marks · Long answerHOTS / competency
Find the smallest 4-digit perfect square that is divisible by each of 6, 8 and 12.
Hint
The LCM is 24; which squares are multiples of 24?
Answer
LCM $= 24 = 2^{3} \cdot 3$ . A square multiple needs at least $2^{4} \cdot 3^{2} = 144$ , so squares are $(12 k)^{2}$ . Smallest 4-digit: $k = 3$ gives $3 6^{2} = 1296$ ( $k = 2$ gives $576$ , only 3 digits). Check: $1296 \div 24 = 54$ .

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.When is a number a perfect square in terms of its prime factors?Show answer
When every prime appears an even number of times.
2.What is the square root of 3600?Show answer
60.
3.Can a perfect square end in 7?Show answer
No.
4.What do you multiply 12 by to make a perfect square?Show answer
3 (giving 36).

Square Root by Prime Factorisation: frequently asked questions

Why does pairing work?

Because \sqrt{p \cdot p} = p; each pair of equal primes becomes one prime in the root.

Does this method work for decimals?

Only after writing the decimal as a fraction; the long division method is better for decimals.