Question

Let s be the sum of ten consecutive integers and let p be their product. Explain why the units digit of P-S must be 5.

Answer 1

In any string of consecutive integers, there must appear a 2, a 5 and a 0,

so the product of any 10 consecutives must end in ...00

let the first of the number be a
The the sum of 10 cosecutives
= (10/2)(2a + 9(1)) = 10a + 45

10a must end in a 0, so subtracting 10 from the product which ends in at least 00 will keep the last digit as a 0
subtracting 45 from last will leave us with a 5 in the unit digit.