Question

If n is an integer then, find the value of(2n-1)"mod 16
(a) 1
(b) 0
(c) 2
(d) 3​

Answer 1

Given :  n is an integer

To Find :  value of(2n-1)⁴ mod 16

(a) 1

(b) 0

(c) 2

(d) 3​

Solution:

(2n-1)⁴ mod 16

(2n-1)⁴  =  ⁴C₀(2n)⁴(-1)⁰  + ⁴C₁(2n)³(-1)¹  +  ⁴C₂(2n)²(-1)²  + ⁴C₃(2n)¹(-1)³ +  ⁴C₄(2n)⁰(-1)⁴

= 16n⁴  + 4(8n³)(-1)  + 6(4n²) + 4(2n)(-1)  + 1

=  16n⁴  - 32n³ + 24n²  - 8n  + 1

=  16n⁴ - 32n³ + 16n² + 8n² - 8n + 1

= 16n²(n²  - 2n + 1)  + 8n(n - 1) + 1

= 16n²(n - 1)² + 8n(n - 1)  +  1

n(n - 1)  = 2k

= 16n²(n - 1)² + 8.2k  +  1

= 16n²(n - 1)² +16k  +  1

= 16(n²(n - 1)² + k) + 1

(2n-1)⁴ mod 16  = 1

the value of (2n-1)⁴ mod 16  = 1

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