Problem of the Week #66: Monday October 7th, 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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Consider a natural number N and the sum of its digits S. Let’s call a divisor of S a Digit Sum Divisor of N.
a). Show that if N is a prime number larger than 10, then N and S are relatively prime.
b). Find all natural numbers k such that if k is a Digit Sum Divisor of both M and N, then k is a Digit Sum Divisor of M+N.
c). Find (with proof) all natural numbers M and N such that M+N = 100, and both M and N have 4 as a Digit Sum Divisor.
d). Find (with proof) all natural numbers M and N such that M-N = 100, and both M and N have 8 as a Digit Sum Divisor.
e). Find all pairs of 1 digit number M and 2 digit number N such that both N and MxN have 5 as a Digit Sum Divisor.
f). Find all pairs of 1 digit number M and multi-digit number N such that both N and MxN have 5 as a Digit Sum Divisor.
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.