SOLUTION: How many positive integers under 1000 with units digit 9 can be expressed as the sum of a power of 2 and a power of 3? Note that 1 is both a power of 2 and a power of 3.

Question 1133240: How many positive integers under 1000 with units digit 9 can be expressed as
the sum of a power of 2 and a power of 3? Note that 1 is both a power of 2 and a power of 3.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The powers of 2 less than 1000 are

1, 2, 4, 8, 16, 32, 64, 128, 256, 512

The powers of 3 less than 1000 are

1, 3, 9, 27, 81, 243, 729

To get an integer with units digit 9 that is the sum of a power of 2 and a power of 3, we can have...

(1) final digits 2 and 7

3 choices for the power of 2 and 1 choice for the power of 3, so 3 answers:
2+27 = 29; 32+27 = 59; 512+27 = 539

(2) final digits 8 and 1

2 choices for the power of 2 and 2 choices for the power of 3, so 4 answers:
8+1 = 9; 8+81 = 89; 128+1 = 129; 128+81 = 209

(3) final digits 6 and 3

2 choices for the power of 2 and 2 choices for the power of 3, so 4 answers:
16+3 = 19; 16+243 = 259; 256+3 = 259; 256+243 = 499

ANSWER: There are 11 positive integers less than 1000 that can be expressed as the sum of a power of 2 and a power of 3.
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