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Problem 254

AMC 10/12, early questions

Algebra Difficulty 4.0 Find the answer South African Mathematics Olympiad Second Round · South Africa

1. Joyce has a box with light bulbs, some blue and some red. She counts the bulbs and finds that 10 of the 40 are blue. What percentage of the bulbs are blue?

2. If $2016 \times x$ is a perfect square, what is the smallest positive integer value for $x$ ?

Your answer

Official solution

1.
$\frac{10}{40} \times 100 = 25.$
So, $25\%$ of the bulbs are blue.

2.
The prime factorisation of $2016$ is $2^5 \times 3^2 \times 7$ . In a perfect square, each prime factor has an even exponent, so we need one more power of $2$ and one more power of $7$ . The answer is $2 \times 7 = 14$ .

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