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Totient maximum
Problem 69
Published on 07 May 2004 at 06:00 pm [Server Time]
Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of numbers less than n which are relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, φ(9)=6.
| n | Relatively Prime | φ(n) | n/φ(n) |
| 2 | 1 | 1 | 2 |
| 3 | 1,2 | 2 | 1.5 |
| 4 | 1,3 | 2 | 2 |
| 5 | 1,2,3,4 | 4 | 1.25 |
| 6 | 1,5 | 2 | 3 |
| 7 | 1,2,3,4,5,6 | 6 | 1.1666... |
| 8 | 1,3,5,7 | 4 | 2 |
| 9 | 1,2,4,5,7,8 | 6 | 1.5 |
| 10 | 1,3,7,9 | 4 | 2.5 |
It can be seen that n=6 produces a maximum n/φ(n) for n ≤ 10.
Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum.
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