Suppose x, y are integers and x mod 12 = 9 and y mod 6 = 2,Find(i) x mod 6(ii) xy mod 6
Question
Suppose x, y are integers and x mod 12 = 9 and y mod 6 = 2,Find(i) x mod 6(ii) xy mod 6
Solution
(i) x mod 6
Given that x mod 12 = 9, we know that x can be expressed as 12n + 9 for some integer n.
When we take x mod 6, we are essentially finding the remainder when x is divided by 6.
So, we need to find the remainder when (12n + 9) is divided by 6.
Since 12n is divisible by 6 (as 12n = 6*2n), it leaves no remainder.
Therefore, the remainder when x is divided by 6 is the same as the remainder when 9 is divided by 6, which is 3.
So, x mod 6 = 3.
(ii) xy mod 6
We know that x mod 6 = 3 and y mod 6 = 2.
When we multiply x and y, the result mod 6 is the same as (x mod 6 * y mod 6) mod 6.
So, xy mod 6 = (3 * 2) mod 6 = 6 mod 6 = 0.
So, xy mod 6 = 0.
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