Problem of the Week #68: Monday October 21st, 2024
As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems.
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This problem is inspired by Euclid 1998 #10.
The number 6 has two pairs of factors: 2×3 and 1×6. These two pairs of factors are called friendly because the sum of one pair is equal to the difference of the other pair, ie. 2+3 = 6-1 = 5.
a). Find another number that has two pairs of friendly factors.
b). Are there infinitely many numbers with friendly factor pairs?
c). Does a number have to be a multiple of 6 to have friendly factor pairs?
d). Find a set of friendly factor pairs that are NOT a multiple of the original example.
e). In the original example, let’s call 5 the friendly sum. Are all friendly sums a multiple of the biggest number in a Pythagorean Triple?
f). Can the biggest number in any Pythagorean Triple be friendly sums?
g). Share your own problem inspired by this one.
h). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.