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June 16, 2004
Powers of Five
What's the remainder when you divide the number 5999,000 by 7?
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You don't need to multiply out the number. Look for a pattern in the remainders you get after dividing powers of 5 by 7.
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The remainder is 1.
The remainder of powers of 5 after dividing by 7 repeats in cycles of six: 5, 4, 6, 2, 3, 1.
Power |
Value |
Remainder |
1 |
5 |
5 |
2 |
25 |
4 |
3 |
125 |
6 |
4 |
625 |
2 |
5 |
3,125 |
3 |
6 |
15,625 |
1 |
7 |
78,125 |
5 |
8 |
390,625 |
4 |
9 |
1,953,125 |
6 |
10 |
9,765,625 |
2 |
Divide 999,000 by 6. There's no remainder. So, the remainder when 5999,000 is divided by 7 is the same as that for 56: the sixth member of the cycle, 1.
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