Which of the following best approximates the… | GMAT Sprint

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PS · Problem Solving

705-805

Divisibility/Multiples/Factors

Which of the following best approximates the value of n if $5^{22} = 10^n$ ?

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Official Explanation

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We need to approximate n such that

5^{22} = 10^n

.

Rewrite

10^n

(5 \times 2)^n = 5^n \times 2^n

.

So the equation becomes

5^{22} = 5^n \times 2^n

.

Divide both sides by

5^n

5^{22-n} = 2^n

.

We want n such that both sides are roughly equal. Taking logs or testing options:

Test n = 15:

Left side:

5^{22-15} = 5^7

.

Right side:

2^{15}

.

Now

5^7 = 78125

and

2^{15} = 32768

.

Ratio:

\frac{5^7}{2^{15}} \approx \frac{78125}{32768} \approx 2.38

. This is close to 2.…

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What this question tests

Your understanding of divisibility rules, least common multiples, and greatest common factors.

Forgetting that 1 is not prime and that every integer divides 0.