Puzzles
20 December
A number is called a perfect power if it is equal to nk for some integer n and some integer k > 1. 2025 is a perfect power (452)
and 23 more than 2025 is also a perfect power (211).
What is the only three-digit perfect power that is 29 less than another perfect power?
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29 is odd, and the difference between consecutive square numbers is always odd so it makes sense to check if the relevant two consecutive squares have three digits: these squares are 196 and 225.
14 December
153 is 3375. The last 3 digits of 153 are 375.
What are the last 3 digits of 151234567890?
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We can look at the first few powers and look for a pattern:
| n | 15n | Last 3 digits of 15n |
| 1 | 15 | 15 |
| 2 | 225 | 225 |
| 3 | 3375 | 375 |
| 4 | 50625 | 625 |
| 5 | 759375 | 375 |
| 6 | 11390625 | 625 |
| 7 | 170859375 | 375 |
| 8 | 2562890625 | 625 |
| 9 | 38443359375 | 375 |
If n is even (and > 2), the last three digits of 15n are 625.
