Let a, b … {1, 2, 3, 4, 5, 6, 7, 8, 9}.
How many pairs (a,b) are there, such that a123 + b456 is divisible by 5 ?
To decide if a number is divisible by 5, we only need to consider the last digit – if the last digit is a 0 or 5 then the number is divisible by 5.
We can consider the last digit as we go through the powers of that number. For example, when considering the powers of 2:
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…
so the last digit repeats the cycle 2, 4, 8, 6. When we consider this for each of the numbers 1 to 9 , we get the following cycles:
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