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The sum of an odd and even number is always odd because there's always one left over. Think of any odd number as being any even number plus one. If you add two even numbers, the answer is always even. If you add an even and an odd (plus one), the answer is always odd.
62 = 2*31
3 + 5 + 13 + 17 + 19 + 23 = 80 3 x 5 x 13 x 17 x 19 x 23 = 1,448,655
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