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#57 Prime jumps

set by Christopher Dearlove

If we start with 21 (which isn’t a prime number), how many times do we have to add 100 before we reach a prime number?

Which numbers below 100 can never be made to reach a prime number by adding multiples of 100?

Returning to the case of 21: one other non-prime below 100 needs the same multiple of 100 added to it to reach a prime. What is it?

Solution next week

#56 Square sums by four

Solution

The first number is 7, for which 1₂ + 2₂ + … + 7₂ is 140.

The remainder (on division by 4) of 1₂ + 2₂ + … + 20₂ = 2870 is 2.

The remainders will follow the pattern 1, 1, 2, 2, 3, 3, 0, 0, then repeat. Adding the square of an even number doesn’t change the remainder on division by 4, since the even number is a multiple of 2, and so its square is a multiple of 4.

Adding the square of an odd number will increase the remainder by 1, since odd numbers are of the form 2n+1 and their squares are therefore 4n₂ + 4n + 1, which is one more than a multiple of 4.