When the positive integer N is divided by 17, the remainder is four times as much as when N is divided by 13.
Find the least possible value that N could represent.
No computer needed.
The pairs of remainders can be any of (4,1), (8,2), (12,3), (16,4)
There are four cases:
i) N = 17j + 4 = 13k + 1 --> 13k - 17j = 3
ii) N = 17j + 8 = 13k + 2 --> 13k - 17j = 6
iii) N = 17j + 12 = 13k + 3 --> 13k - 17j = 9
iv) N = 17j + 16 = 13k + 4 --> 13k - 17j = 12
(and 17j is always 0 mod 17)
Starting from k = 0 then incrementing 1, the mod 17 value of 13k goes through a particular sequence where you either add 13 or subtract 4:
0, 13, 9, 5, 1, 14, 10, 6, 2, 15, 11, 7, 3, 16, 12, 8, 4, 0
We are looking for whichever case shows up first with a value of 3, 6, 9, or 12.
The first to show up is 9, (case iii) corresponding to k = 2, so N = 13*2 + 3 = 29
N = 29
29/17 = 1 12/29
29/13 = 2 3/29
The remainder 12 is 4 times the remainder 3.
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Posted by Larry
on 2024-02-15 11:51:18 |
Analytic solution
