The function f is defined for each positive three-digit integer n by , where x, y and z are the hundreds, tens, and units digits of n, respectively. If m and v are three-digit positive integers such that , then ?
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This one took me a while to wrap my head around. The trick for me was realizing that multiplying by 9 = 3^2 just bumps up the y exponent by 2. Once I saw it that way, the digit relationship became clear.
Same here, the 9 = 3^2 insight was the whole game. Before that I was trying to brute-force values and getting nowhere.
Wait, so f(m) = 9f(v) means 2^x 3^y 5^z = 9 · 2^a 3^b 5^c. Does that mean x = a and z = c for sure? Like, why can't the factors redistribute?
Right, unique prime factorization forces each prime's exponent to be equal on both sides.
Honestly the hardest part was remembering x,y,z are digits of a three-digit number, not just any variables. The setup is sneaky.
Tough question, felt more like 700+ to me. Got there eventually but spent way too long second-guessing my digit assignments.
What it tests
Your command of divisibility, primes, factors, parity, and how the GMAT tests properties of integers.
Common trap
Forgetting that 1 is not prime, that 0 is divisible by every integer, or that negative numbers flip your assumptions.