Suat AyözProblem for March 19th from the 2026 AMS Daily Epsilon of Math Calendar
Alex Bolton
I thought
10x = 646 – 95z – 38y = 19•(34 – 5z – 2y)
Hence, 10x must be a multiple of 19. 10 isn't, so x must be. But x is a prime, so the only number that is prime and a multiple of 19 is 19 itself.
bdesmons@PercyButtons3 @DailyEpsilon That's the most direct way indeed ; but let's mention Gauss' theorem to state that, when 10x is a multiple of 19, then so is x ;)
I first thought about a similar solution as @alexdbolton's: observe that z must be even, so z=2, and solve \( 10 x + 38y = 646 - 190 = 456\). Divide everything by 2 to get \(5x +19y=228 \). As 228 is not a multiple of 5, I thought : is it a multiple of 19? Splitting 228 = 190+38 showed me it was! So I concluded :)
I first thought about a similar solution as @alexdbolton's: observe that z must be even, so z=2, and solve \( 10 x + 38y = 646 - 190 = 456\). Divide everything by 2 to get \(5x +19y=228 \). As 228 is not a multiple of 5, I thought : is it a multiple of 19? Splitting 228 = 190+38 showed me it was! So I concluded :)