2008 iTest Problems/Problem 39
Problem
Let denote , the number of integers that are relatively prime to . (For example, and .) Let , in which ranges through all positive divisors of , including and . Find the remainder when is divided by .
Solution
The divisors of are ,,,,,,, and , so the problem requires the sum of the number of numbers relatively prime to each of the eight numbers.
Using the formula (or by manually counting the numbers relatively prime to each number), number is relatively prime to , number is relatively prime to , numbers are relatively prime to , numbers are relatively prime to , numbers are relatively prime to , numbers are relatively prime to , numbers are relatively prime to , and numbers are relatively prime to . That means , and the remainder when is divided by is .
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 38 |
Followed by: Problem 40 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • 61 • 62 • 63 • 64 • 65 • 66 • 67 • 68 • 69 • 70 • 71 • 72 • 73 • 74 • 75 • 76 • 77 • 78 • 79 • 80 • 81 • 82 • 83 • 84 • 85 • 86 • 87 • 88 • 89 • 90 • 91 • 92 • 93 • 94 • 95 • 96 • 97 • 98 • 99 • 100 |