2012 UNCO Math Contest II Problems/Problem 1

Problem

(a) What is the largest factor of $180$ that is not a multiple of $15$ ?

(b) If $N$ satisfies $2^5 \times 3^2 \times5^4 \times 7^3 \times 11=99000 \times N$ , then what is the largest perfect square that is a factor of $N$ ?

Solution

(a)  (b)

$(a)$

The prime factorization of $180$ is

$2^2 \times 3^2 \times5$ .

We need to get the largest factor of $180$ which isn't a multiple of $15$ , so we can't include both $3$ and $5$ . $3^2$ is greater than $5$ , so we'll use that and $2^2$ to get $3 \times 3 \times 2 \times 2 = \fbox{\textbf 36}$ .

$(b)$

$2^5 \times 3^2 \times5^4 \times 7^3 \times 11=99000 \times N$

$2^5 \times 3^2 \times5^4 \times 7^3 \times 11=2^3 \times 3^2 \times 5^3 \times 11 \times N$

$2^2 \times 5 \times 7^3= N$

The largest square factor of $N$ is $2^2$ $\times$ $7^2$ = $\fbox{\textbf 196}$

2012 UNCO Math Contest II (Problems • Answer Key • Resources)
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2012 UNCO Math Contest II Problems/Problem 1

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