Problem 27 · 2024 Math Kangaroo
Stretch
Number Theory
divisibility
A farmer sells chicken eggs and duck eggs. In different baskets he has 4, 6, 12, 13, 22 and 29 eggs, with each basket holding eggs of only one kind. Christoph buys all the eggs in one basket. The farmer then realises that he now has twice as many chicken eggs as duck eggs. How many eggs did Christoph buy?
Show answer
Answer: E — 29
Show hints
Hint 1 of 2
After Christoph takes one basket, the rest must split as chicken = twice duck, so the remaining total is a multiple of 3.
Still stuck? Show hint 2 →
Hint 2 of 2
Find which single basket, when removed, leaves a multiple of 3.
Show solution
Approach: use divisibility by 3 on the remaining total
- All baskets total 4+6+12+13+22+29 = 86.
- Remaining = chicken + duck = 2·duck + duck = 3·duck, so it must be a multiple of 3.
- 86 leaves remainder 2 mod 3, so the removed basket must be ≡ 2 mod 3; only 29 qualifies.
- Christoph bought the basket of 29 eggs.
Mark:
· log in to save