Which of the following numbers is not a facto | vedclass

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(D) number $x$ is a factor of $5^{p} 7^{q}$ if and only if the prime factorization of $x$ contains only the primes $5$ and $7$,with exponents less tha

Vedclass · Accepted Answer

$735$

Vedclass · Answer

(D) number $x$ is a factor of $5^{p} 7^{q}$ if and only if the prime factorization of $x$ contains only the primes $5$ and $7$,with exponents less than or equal to $p$ and $q$ respectively.Given the expression $5^{p} 7^{q}$,any factor must be of the form $5^{a} 7^{b}$ where $0 \leq a \leq p$ and $0 \leq b \leq q$.Let's analyze the prime factorization of the given options:$A) 35 = 5^1 	imes 7^1$. This is a factor if $p \geq 1$ and $q \geq 1$.$B) 175 = 5^2 	imes 7^1$. This is a factor if $p \geq 2$ and $q \geq 1$.$C) 1225 = 5^2 	imes 7^2$. This is a factor if $p \geq 2$ and $q \geq 2$.$D) 735 = 5^1 	imes 7^2 	imes 3^1$. Since $735$ contains a prime factor $3$,which is not present in the expression $5^{p} 7^{q}$,it can never be a factor regardless of the values of $p$ and $q$ (provided $p, q 
eq 0$ as given).Therefore,$735$ is not a factor.

Which of the following numbers is not a factor of $5^{p} 7^{q}$ where $p \neq 0$ and $q \neq 0$?

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