The number of positive integers less than 50, having exactly two distinct factors other than 1 and itself, is

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15 — Easy "Exactly two distinct factors other than $1$ and itself" means exactly $4$ total factors. A number has exactly $4$ factors only when it is $p\cdot q$ (two distinct primes) or $p^3$ (cube of a prime). Count both families below $50$. Number of factors of $p^a q^b\cdots$ is $(a+1)(b+1)\cdots$. We need this to equal $4$, so the exponent pattern is either $1	imes1$ → $p^1q^1$, or $3$ → $p^3$. 1 Case A — product of two distinct primes $p\cdot q\lt 50$: $$\begin{aligned} &p=2:\ q\in\{3,5,7,11,13,

CAT 2023 Slot 2 — QA Question 6

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