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

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GMAT Panda · Accepted Answer

5 We first factor out the common 5! from both terms to simplify the expression. 10! - 2(5!)^2 = 5! (6 7 8 9 10 - 2 5!) We then multiply 6 and 7 inside the parentheses. 6 7 = 42 We then multiply the result by 8. 42 8 = 336 We simplify 336 9 by distributing to avoid large-factor multiplication. 336 = 300 + 36 300 9 = 2700 36 9 = 324 2700 + 324 = 3024 We then multiply the result by 10. 3024 10 = 30240 We then subtract twice 5! from that result to complete the parentheses. 2 5! = 2 120 = 240 30240 - 240 = 30000 This means the expression becomes 5! times 30000. 5! 30000 = 120 30000 We break each factor into its prime factors to find the exponent of 5 in the product. 5! = 2 3 4 5 = 2 3 (2 2) 5 = 2^3 3 5 30000 = 3 10^4 = 3 (2 5)^4 = 3 2^4 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 3 5 2^4 3 5^4 = 2^7 3^2 5^5

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

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