1, 3, 7, 17, 21, 51, 119, 357
If x ≡ 39 mod 357 then: x = 357k + 39 for some integer k. Now 357 = 21×17, and 39 = 2×17+5 → x = 21×17×k + 2×17 + 5 → x = 17(21k + 2) + 5 → x = 17m + 5 where m = 21k + 2 (is an integer) → x ≡ 5 mod 17 → the remainder when the number is divided by 17 is 5.
23.8
357 divided by 9 is 39.66666667
59.5
1, 3, 7, 17, 21, 51, 119, 357
If x ≡ 39 mod 357 then: x = 357k + 39 for some integer k. Now 357 = 21×17, and 39 = 2×17+5 → x = 21×17×k + 2×17 + 5 → x = 17(21k + 2) + 5 → x = 17m + 5 where m = 21k + 2 (is an integer) → x ≡ 5 mod 17 → the remainder when the number is divided by 17 is 5.
23.8
14 divided into 357 = 0.0392156862745098
89.25
357 divided by 9 is 39.66666667
59.5
1.4691
119
340 = 20 x 17 357 - 340 = 17 17 times with 17 remaining
357 = 3*7*17
If x ≡ w mod 357 then: x = 357k + w for some integer k. Now 357 = 21×17, and w = 17n + c for some integers n ≥ 0 and 0 ≤ c < 17 → w ≡ c mod 17 This gives: x = 21×17×k + 17n + c → x = 17(21k + n) + c → x = 17m + c where m = 21k + n (is an integer) → x ≡ c mod 17 → the remainder when the number is divided by 17 is the same as the remainder when the original remainder w is divided by 17.