Is the sum of four even numbers always a multiple of 4?

Answer 1

No, the sum of four even numbers is not always a multiple of 4.. For example, 4 + 6 + 10 + 14 = 34 which is not a multiple of 4

If you include an "odd" number of even numbers which are not multiples of 4, then the total will not divide exactly by 4.

In the above example 6 = 4 + 2, 10 = 4 + 4 + 2, and 14 = 4 + 4 + 4+ 2, and the total of the odd number of twos will not divide by 4

All multiples of 4 are the product of 2 with another even number:

(Let a multiple of 4 be 4k. But 4 = 2 x 2, so 4k = (2 x 2)k = 2 x 2k and 2k is an even number for all k.)

Let the four even numbers be 2k, 2m, 2n, 2p for some k, m, n, p.

Then their sum is 2k + 2m + 2n + 2p = 2 (k + m + n + p) = 2q where q = k + m + n + p.

k, m, n & p can be even or odd. If one, or three, of these is odd, then their sum, q, is odd.

To be a multiple of 4 the sum k + m + n + p (= q) must be even. If it is odd, then twice it will not be a multiple of 4.

For example, if k = 1, m = 2, n = 3, p = 5, then the four even numbers are 2 x 1 = 2, 2 x 2 = 4, 2 x 3 = 6, 2 x 5 = 10 and their sum is 2 x (1 + 2 + 3 + 5) = 2 x 11 = 22 which is not a multiple of 4.