The number of positive even divisors of 6300 | vedclass

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Question

(D) First,find the prime factorization of $6300$: $6300 = 63 	imes 100 = (9 	imes 7) 	imes (10^2) = (3^2 	imes 7^1) 	imes (2^2 	imes 5^2) = 2^2

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$36$

Vedclass · Answer

(D) First,find the prime factorization of $6300$: $6300 = 63 	imes 100 = (9 	imes 7) 	imes (10^2) = (3^2 	imes 7^1) 	imes (2^2 	imes 5^2) = 2^2 	imes 3^2 	imes 5^2 	imes 7^1$. Total number of divisors is given by the product of (exponent $+ 1$) for each prime factor: Total divisors $= (2+1)(2+1)(2+1)(1+1) = 3 	imes 3 	imes 3 	imes 2 = 54$. Odd divisors are formed by the prime factors excluding $2$: Odd divisors $= (2+1)(2+1)(1+1) = 3 	imes 3 	imes 2 = 18$. Therefore,the number of even divisors is the total number of divisors minus the number of odd divisors: Even divisors $= 54 - 18 = 36$.

The number of positive even divisors of $6300$ is

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