The function f is defined as follows:
For any 3 digit integer (written xyz), f(xyz)=2^x3^y5^z.
If c and k are 3 digit integers, and f(c)=16*f(k), what is the value of c - k?
(A) 400
(B) 320
(C) k/16c
(D) 40
(E) Cannot be determined
For every 3-digit integer xyz, f(xyz) = (2^x)(3^y)(5^z).
Let f(k) = 100.
Thus:
f(xyz) = 100.
Substituting f(xyz) = (2^x)(3^y)(5^z) and prime-factorizing the righthand side, we get:
(2^x)(3^y)(5^z) = 2²3�5².
Since the bases on each side of the equation match, so must the corresponding exponents.
Thus, x=2, y=0, z=2.
Thus, k = (3-digit integer xyz) = 202.
Since f(c) = 16f(k) and f(k)=100, f(c) = 16(100).
Thus:
f(xyz) = 16(100).
Substituting f(xyz) = (2^x)(3^y)(5^z) and prime-factorizing the righthand side, we get:
(2^x)(3^y)(5^z) = (2�)(2²3�5²) = 2�3�5².
Since the bases on each side of the equation match, so must the corresponding exponents.
Thus, x=6, y=0, z=2.
Thus, c = (3-digit integer xyz) = 602.
Thus, c-k = 602-202 = 400.
The correct answer is
A.
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