GMAT Number Properties: What is the greatest positive…

What is the greatest positive integer n such that 5^{n} divides 10! - (2) (5!)^{2} ?

factorialsdivisibilitynumber theoryprime factorizationexponents

Explanation	Calculations	Help
$We first factor out the common 5! from both terms to simplify the expression.$	$10! - 2 (5!)^{2} = 5! \times (6 \times 7 \times 8 \times 9 \times 10 - 2 \times 5!)$
$We then multiply 6 and 7 inside the parentheses.$	$6 \times 7 = 42$
$We then multiply the result by 8.$	$42 \times 8 = 336$
$We simplify 336 \times 9 by distributing to avoid large-factor multiplication.$	$336 = 300 + 36 300 \times 9 = 2700 36 \times 9 = 324 2700 + 324 = 3024$
$We then multiply the result by 10.$	$3024 \times 10 = 30240$
$We then subtract twice 5! from that result to complete the parentheses.$	$2 \times 5! = 2 \times 120 = 240 30240 - 240 = 30000$
$This means the expression becomes 5! times 30000.$	$5! \times 30000 = 120 \times 30000$
$We break each factor into its prime factors to find the exponent of 5 in the product.$	$5! = 2 \times 3 \times 4 \times 5 = 2 \times 3 \times (2 \times 2) \times 5 = 2^{3} \times 3 \times 5 30000 = 3 \times 1 0^{4} = 3 \times (2 \times 5)^{4} = 3 \times 2^{4} \times 5^{4}$
$We then combine the prime factors and apply Divisibility via Prime Exponents to find the largest power of 5 dividing the product.$	$2^{3} \times 3 \times 5 \times 2^{4} \times 3 \times 5^{4} = 2^{7} \times 3^{2} \times 5^{5}$

5 (D)

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