How do I convince my grandmother (who really hates mathematics) that there exist three positive integers $x,y,z$ that satisfy the equation $28x+30y+31z=3650$?
You tell your Grandmother:
Take ten ordinary years. They contain exactly 3650 days, and these days are split among several months with 28, 30, and 31 days.
Your grandmother can probably
answer the question for 365 (days),
and she can probably
multiply by 10.
It's obviously the number of days in 10 non-leap years. and x, y, and z are 10, 40, and 70.