Number theory › Remainders › Numbers with given remainders

Smallest n with fewer pencils after a gift

High school · ●●●○○ 50 pts

I have two cents and Bill has

n

cents. Bill wants to buy pencils, which come in two packages: one package costs 6 cents for 7 pencils, and the other costs 10 cents for 12 pencils. Bill can spend all

n

of his cents on some combination of these packages to get

P

pencils. If I give my two cents to Bill, he can instead spend all

n+2

of his cents on some combination of these packages to get fewer than

P

pencils. Find the smallest value of

n

for which this is possible.

Diophantine equations optimization over integers