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Consider the expression, get zero remainder (Posted on 2007-05-14) Difficulty: 2 of 5
Let q be a positive whole number.

Determine whether or not 1q + 2q + 3q + 4q is always divisible by 10 whenever q is NOT divisible by 4.

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution another way of getting there Comment 2 of 2 |
If q is not divisible by 4, it can be written as (4*k + r), with r in {1,2,3}.
The expression becomes 1 + 2^(4*k+r) + 3^(4*k+r) + 4^(4*k +4). We have to prove this is equivalent to 0 when calculating modulo 10. The totient of 10 is (5-1)x(2-1) = 4 (very convenient).
This means that our expression, when calculating modulo 10 is equivalent to 1 + 2^r + 3^r + 4^r. Checking for all three possible values of r gives the correct result of 0. For r=0, it is equivalent to 4, so for q divisible by 4, the number will always end with a 4.

  Posted by Robby Goetschalckx on 2007-05-14 20:00:52
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