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Definitions
Division algorithm Let a be an integer and d a positive integer. Then there are nique integers q and r with 0 <= r < d such that a=dq+r
q is called the quotient and r is called the remainder
q=a div d
r=a mod d
Solution
3 mod 7
Since 37 is larger than 3, we should be able to obtain 37 by consecutively 7 to 3 if 37 ≡ 3 mod 7.
3 mod 7
≡ 3+7 mod 7
≡ 10 mod 7
≡ 10+7 mod 7
≡ 17 mod 7
≡ 17+7 mod 7
≡ 24 mod 7
≡ 24+7 mod 7
≡ 31 mod 7
≡ 31+7 mod 7
≡ 38 mod 7
We then note that 3 mod 7 is ≡alent with 31 and 38. 3 mod 7 is then not ≡alent with 37, since 31<37<38.
Note 37 mod 7 ≡ 2 mod 7