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Responding to a pm
\(m = 3^{x-y}*5^{2y-1}*7^{4-x}\)
\(n = 105^y = 3^y * 5^y * 7^y\)
What is y?
(1) n is not a multiple of m
Many cases are possible.
y could be 0. Then n = 1.
If say x = 1 then m = 3*7^3/5
y could be 1. Then n = 105.
If say x = 1 then m = 5*7^3
Not sufficient alone.
(2) m is a multiple of n.
Since m is a multiple of n and multiples are defined for integers only, m and n must be integers. Since m is a multiple of n, its exponents of all primes must be greater than or equal to exponents of n.
\(x - y \geq y\) which gives \(x \geq 2y\) -------(1)
\(2y - 1 \geq y\) which gives \(y \geq 1\)
\(4-x \geq y \) which gives\( - x \geq y - 4\) ------(3)
Adding (1) and (3)
\(0 \geq 3y-4\)
\(y \leq 4/3\)
Since y must be an integer, y can only be 1.
Answer (B)
\(m = 3^{x-y}*5^{2y-1}*7^{4-x}\)
\(n = 105^y = 3^y * 5^y * 7^y\)
What is y?
(1) n is not a multiple of m
Many cases are possible.
y could be 0. Then n = 1.
If say x = 1 then m = 3*7^3/5
y could be 1. Then n = 105.
If say x = 1 then m = 5*7^3
Not sufficient alone.
(2) m is a multiple of n.
Since m is a multiple of n and multiples are defined for integers only, m and n must be integers. Since m is a multiple of n, its exponents of all primes must be greater than or equal to exponents of n.
\(x - y \geq y\) which gives \(x \geq 2y\) -------(1)
\(2y - 1 \geq y\) which gives \(y \geq 1\)
\(4-x \geq y \) which gives\( - x \geq y - 4\) ------(3)
Adding (1) and (3)
\(0 \geq 3y-4\)
\(y \leq 4/3\)
Since y must be an integer, y can only be 1.
Answer (B)
