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CAT 2019 Slot 2 — QA Question 6

FactorsEasy

How many factors of 24 × 35 × 104 are perfect squares which are greater than 1?

Answer & solution

Correct answer: 44

Answer: 44

Solution

Easy

Write the number in prime-factored form first. A factor is a perfect square exactly when every prime's exponent is even, so count the even choices for each prime independently, multiply, then drop the factor 11.

1

Prime factorise. Note 104=(2⋅5)4=24⋅5410^{4}=(2\cdot 5)^{4}=2^{4}\cdot 5^{4}.

N=24×35×104=24×35×24×54⇒ N=28×35×54\begin{aligned} &N=2^{4}\times 3^{5}\times 10^{4}=2^{4}\times 3^{5}\times 2^{4}\times 5^{4}\\ &\Rightarrow\ N=2^{8}\times 3^{5}\times 5^{4} \end{aligned}
2

Count even exponents for each prime. A square factor needs an even power of every prime.

28: {0,2,4,6,8} → 5 choices35: {0,2,4} → 3 choices54: {0,2,4} → 3 choices\begin{aligned} &2^{8}:\ \{0,2,4,6,8\}\ \to\ 5\ \text{choices}\\ &3^{5}:\ \{0,2,4\}\ \to\ 3\ \text{choices}\\ &5^{4}:\ \{0,2,4\}\ \to\ 3\ \text{choices} \end{aligned}
3

Multiply and exclude 11. The all-zero choice gives the factor 11, which the question excludes.

Square factors=5×3×3=45⇒ greater than 1=45−1=44\begin{aligned} &\text{Square factors}=5\times 3\times 3=45\\ &\Rightarrow\ \text{greater than }1=45-1=44 \end{aligned}
4444

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CAT 2019 Slot 2 QA Q6: How many factors of 2 4 × 3 5 × 10 4 are perfect squares which are greater than 1? — Solution | TheCATExam