For all integers a, b, c, m with m > 0 and c > 0, if ac ≡ bc (mod m), then a ≡ b (mod m
Question
For all integers a, b, c, m with m > 0 and c > 0, if ac ≡ bc (mod m), then a ≡ b (mod m
Solution
The statement is true. Here's why:
The congruence ac ≡ bc (mod m) means that m divides ac - bc.
We can factor out c from the expression to get c(a - b). So, m divides c(a - b).
Since m > 0 and c > 0, we know that the greatest common divisor (gcd) of m and c is a positive integer.
If the gcd of m and c is 1, then m and c are coprime. According to a property of coprime numbers, if a product of two numbers is divisible by a coprime number, then the other factor is also divisible by that coprime number.
So, in this case, if m divides c(a - b), and m and c are coprime, then m must also divide (a - b).
Therefore, a ≡ b (mod m).
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