33m - 7n can only equal 26 or values greater than 26. By considering the remainders when dividing 33m - 7n by 16 and 3, it is shown that no smaller values than 26 are possible. Dividing by 16 shows the only possible answers are 0, 10, 16, or 26. Dividing by 3 leaves 26 as the only possibility, since 33m - 7n would have a remainder of 2 when divided by 3 for any other value.

1. Solution
Week 30 (4/7/03)
Difference of Powers
A value of 26 is obtainable with m = n = 1. By considering the remainder when
33m − 7n is divided by certain numbers, we will show that no value smaller than 26
is possible. We will use the “mod” notation for convenience, where a ≡ b (mod c)
means that a leaves a remainder of b when divided by c.
• Consider divisions by 16. We have 33 ≡ 1 (mod 16), and 7n ≡ 7 or 1 (mod
16) because 72 ≡ 1 (mod 16). Therefore, 33m − 7n ≡ 0 or 10 (mod 16). So
the only possible answers to the problem are 0, 10, 16, and 26.
• Now consider divisions by 3. We have 33 ≡ 0 (mod 3), and 7 ≡ 1 (mod 3).
Therefore, 33m − 7n ≡ 2 (mod 3). This leaves 26 as the only possibility.
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