For positive integers x, y, and z, the function x ? y ? z means that

Question

For positive integers x, y, and z, the function x ? y ? z means that \(\frac{z}{x^y}\) is an integer, but \(\frac{z}{x^{y+1}}\) is not. If x ? 2 ? 144, then which of the following is NOT a possible value of x ?

A. 2
B. 3
C. 4
D. 6
E. 12

A. 2
B. 3
C. 4
D. 6
E. 12

Showmeyaa · Answer

So, as per the function: 144/x^2 should be an integer and 144/x^3 shouldn't be an integer.
Let's try plugging in the values from the options:
x = 2, x^2 = 4, x^3 = 8
So, 144/x^2 = 144/4 = 38(Integer)
Also, 144/x^3 = 144/8 = 18(Integer), but as per the condition it shouldn't be an integer.

IMO, (A)!

IanStewart · Answer

It's not correct to describe the notation in the question as a "function" (it doesn't map an input value or set of input values to a well-defined output). It really should just be called a "notation". The answer is 2, since if 2 ? 2 ? 144 were true, that would mean 144 is divisible by 2^2 (and it is) but not by 2^3, but 144 is divisible by 2^3.

jeanslew1 · Answer

Here's what I've got. Following the notation, it suggests that:

\frac{144}{x^{2}}  is an integer

\frac{144}{x^{3}}  is NOT an integer

Breaking down 144 into its prime factors and we get  144 = 2^4(3^2) .

Now, going through the possible answers to check if the number is divisible:
(A)  144/x^2  is an int, as  2^2  exists in 144.  144/2^3  will also be an int, as  2^3  exists in 144. Therefore, 2 cannot be a possible answer as we need the second result to not be an integer.
(B)  144/x^2  is an int, as  3^2  exists in 144.  144/3^3  will NOT be an int, as  3^3  is not part of 144. 3 is a possible value for x.
(C), (D), (E) follow same approach

bumpbot · Answer

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For positive integers x, y, and z, the function x ? y ? z means that

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For positive integers x, y, and z, the function x ? y ? z means that

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