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M = (a + b)^2-ab is a multiple of 5 for even a,b

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2018 3.5

September 20, 2021

number theorydivisibledivides

Problem Statement

Let

a

and

b

be even numbers, such that

M = (a + b)^2-ab

is a multiple of

5

. Consider the following statements: I) The unit digits of

a^3

and

b^3

are different. II)

M

is divisible by

100

. Please indicate which of the above statements are true with certainty.