GMAT Practice Question: The function f is defined for each positive three-digit integer n by...
Question
The function f is defined for each positive three-digit integer n by f(n)=2^{x} \times 3^{y} \times 5^{z}, 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 f(m)=9 f(v), then m-v=
- 8
- 9
- 18
- 20
- 80
Topics: functions, exponents, prime factorization, equations of primes
Solution
Step 1
We let the digits of m be x_m, y_m, and z_m, and the digits of v be x_v, y_v, and z_v.
f(m) = 2^{x_m} \times 3^{y_m} \times 5^{z_m} f(v) = 2^{x_v} \times 3^{y_v} \times 5^{z_v}
Step 2
We express the ratio of f(m) to f(v) in terms of the digit differences.
\frac{f(m)}{f(v)} = 2^{x_m - x_v} \times 3^{y_m - y_v} \times 5^{z_m - z_v}
Step 3
We set this ratio equal to 9 and write 9 as a power of primes.
\frac{f(m)}{f(v)} = 3^{2}
Step 4
We match each prime exponent on both sides to find how the digits differ.
x_m - x_v = 0 y_m - y_v = 2 z_m - z_v = 0
Step 5
We express m - v in terms of the digit differences.
m - v = 100 \times (x_m - x_v) + 10 \times (y_m - y_v) + (z_m - z_v)
Step 6
We substitute the differences and calculate.
m - v = 100 \times 0 + 10 \times 2 + 0 = 20
Answer
20
