Ana has 22 coins. She can take from her friends either 6 coins or 18 coins, or she can give 12 coins to her friends. She can do these operations many times she wants. Find the least number of coins Ana can have.
Problem
Source: Kosovo Mathematical Olympiad 2022, Grade 9, Problem 1
Tags: combinatorics
StarLex1
06.03.2022 23:10
22+18k+6l−12d=0 22+18k+6l−12d=1(not possible since all even) . . . . . 22+18k+6l−12d=n 2m=n we can also say that m=−6d+9k+3l+11 0=−6d+9k+3l+11
l=2d−3k−113 obviously non int since d and k are integer
1=−6d+9k+3l+11
l=2d−3k−103 obviously non int since d and k are integer
2=−6d+9k+3l+11 l=2d−3k−3 for 2d≥3k+3 hence the least number that ana could have is 4 take example (l,d,k)=(0,3,1) 22+18−36=4
hhh1234
07.03.2022 06:59
mod 6 and ez game
Rijadinho
28.04.2023 00:10
Since 6 \equiv 12 \equiv 18 \equiv 0 \pmod{6} And 22 \equiv 4 \pmod{6} It is clear that 4 is the least amount of coins Ana can have \blacksquare