Problem 15 · 2011 AMC 8
Easy
Number Theory
exponent-rewrite
How many digits are in the product 45 · 510?
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Answer: D — 11 digits.
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Hint 1 of 2
Don't multiply it out — every 2 pairs with a 5 to make a 10, so rewrite the whole product as a power of 10.
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Hint 2 of 2
45 is really 210, which gives you exactly ten 2's to match the ten 5's.
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Approach: pair every 2 with a 5 to build a power of 10
- Rewrite 45 as a power of 2: 45 = (22)5 = 210. Now you have ten 2's and ten 5's.
- Each 2 pairs with a 5 to make a 10: 210 · 510 = (2·5)10 = 1010.
- 1010 is a 1 followed by ten zeros — that's 11 digits (the 1 plus 10 zeros, not 10).
- Worth keeping: a stack of 2's times a stack of 5's always collapses into a power of 10. Watch the off-by-one: 10n has n+1 digits.
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