No one is perfect!

Number Theory Level 2

Which of the following numbers is a perfect square ?

101 ! × 99 ! 101!\times 99! None of these choices 100 ! × 101 ! 100!\times 101! 99 ! × 100 ! 99!\times 100! 98 ! × 99 ! 98!\times 99!

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Anish Harsha by Anish Harsha , 20, India

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3 solutions

Akshat Sharda
Apr 28, 2016

n ! × ( n + 1 ) ! n!\times (n+1)! can be a perfect square when n + 1 n+1 is itself a perfect square because n ! × ( n + 1 ) ! = ( n ! ) 2 × ( n + 1 ) n!\times (n+1)!=(n!)^2\times (n+1) .

Therefore, 99 ! × 100 ! 99!\times 100! is perfect square.

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Chew-Seong Cheong
Apr 28, 2016

99 ! × 100 ! = 99 ! × 99 ! × 100 = ( 99 ! ) 2 × 1 0 2 99! \times 100! = 99! \times 99! \times 100 = (99!)^2\times 10^2 , a perfect square.

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Ashish Menon
Jun 5, 2016

99 ! × 100 ! = 99 ! × 99 ! × 100 = ( 99 ! ) 2 × 10 2 = ( 10 × 99 ! ) 2 99! × 100! = 99! × 99! × 100 = {\left(99!\right)}^{2} × {10}^2 = {\left(10 × 99!\right)}^2 .
So, 99 ! × 100 ! \color{#3D99F6}{\boxed{99! × 100!}} is a perfect square.

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