A problem by Siddharth Simharaju

Level pending

The owner of a pet store tried to clean the cages of some canaries, but then when he tried to put them back he was lost. If he puts in one canary per cage, he has one bird without a cage. If he puts in two canaries per cage, he has one cage empty. How many cages does he have?

5 4 2 3

Siddharth Simharaju by Siddharth Simharaju , 45, India

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1 solution

Chew-Seong Cheong
Sep 21, 2015

Let the number of cages be n n , then the number of canaries is n + 1 n+1 . Since n + 1 n+1 is divisible by 2 2 , then n n must be odd and let it be 2 m + 1 2m+1 . Then we have:

n + 1 2 = n − 1 2 m + 1 + 1 2 = 2 m + 1 − 1 2 m + 2 2 = 2 m m + 1 = 2 m ⇒ m = 1 ⇒ n = 2 ( 1 ) + 1 = 3 \begin{aligned} \frac{n+1}{2} & = n - 1 \\ \frac{2m+1+1}{2} & = 2m+1 - 1 \\ \frac{2m+2}{2} & = 2m \\ m+1 & = 2m \\ \Rightarrow m & = 1 \\ \Rightarrow n & = 2(1) + 1 = \boxed{3} \end{aligned}

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