Natural!

Number Theory Level 3

If n is the minimum positive integer that is divisible by the first twenty natural numbers,find:

((n+1)^2)-((n-1)^2)


The answer is 931170240.

Prateek Mehra by Prateek Mehra , 22, India

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2 solutions

Rick B
Jan 9, 2015

n = lcm ( 1 , 2 , 3 , … , 20 ) = 232792560 n = \text{lcm}(1,2,3,\ldots,20) = 232792560

⟹ ( n + 1 ) 2 − ( n − 1 ) 2 = [ ( n + 1 ) + ( n − 1 ) ] × [ ( n + 1 ) − ( n − 1 ) ] ⏟ difference of squares \implies \underbrace{(n+1)^2-(n-1)^2 = [(n+1)+(n-1)] \times [(n+1)-(n-1)]}_{\text{difference of squares}}

= 2 n × 2 = 4 n = 4 × 232792560 = 931170240 = 2n \times 2 = 4n = 4 \times 232792560 = \boxed{931170240}

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Rommel Ocampo
Jan 9, 2015

squaring a binomial n^2+2n+1 -(n^2-2n+1) = 4n n = 16(5)(7)(9)(11)(13)(17)(19)=232792560 4n = 4(232792560)=931170240

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Would please explain??

Anandhu Raj - 6 years, 5 months ago

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