Question

Find the least positive integer n such that 2^2000 is divisible by 1+nc1+nc2+nc3

Answer 1

Heya user,

nC1 = n
nC2 = n(n-1) / 2
nC3 = n(n-1)(n-2) / 6

Now, nC1 + nC2 + nC3 = ( 6n + 3n² - 3n + n³ - 2n² - n² + 2n ) / 6
=> nC1 + nC2 + nC3 = ( n³ + 5n ) / 6
=> nC1 + nC2 + nC3 + 1 = (n³ + 5n + 6) / 6
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Now, 6 | ( n³ + 5n + 6 ) = > 6 | n(n² + 5) => n = 1 satisfies the eqn.

Putting n=1, ( n³ + 5n + 6 ) / 6 = ( 1+5+6 ) / 6 = 2 which divides 2^2000

Hence, n=1 is the least positive integer n such that 2^2000 is divisible by 1+nc1+nc2+nc3