Find-the-number-of-x-1-2016-x-N-which-making-the-expression-4x-6-x-3-5-is-divided-by-11-

Question Number 157599 by naka3546 last updated on 25/Oct/21

F i n d t h e n u m b e r o f x \in [1, 2016], x \in N

w h i c h m a k i n g t h e e x p r e s s i o n 4 x^{6} + x^{3} + 5 i s d i v i d e d b y 11 .

Answered by TheSupreme last updated on 25/Oct/21

4 x^{6} + x^{3} + 5 \equiv 0 (11)

4 x^{6} + x^{3} \equiv 6 (11)

x^{3} (4 x^{3} + 1) \equiv 6 (11)

x^{3} = A

A (4 A + 1) \equiv 6 (11)

A \equiv 1 (11) 5

A \equiv 2 (11) 7

A \equiv 3 (11) 6

A \equiv 4 (11) 2

A \equiv 5 (11) 8

A \equiv 6 (11) 7

A \equiv 7 (11) 5

A \equiv 8 (11) 0

A \equiv 9 (11) 3

A \equiv 1 (11) 0 3

A = 0 (11) 0

\dots

A \equiv 3 (11)

x^{3} \equiv 3 (11)

x \equiv 1 (11) 1

x \equiv 2 (11) 4

x \equiv 3 (11) 9

x \equiv 4 (11) 5

x \equiv 5 (11) 3

x \equiv 6 (11) 3

x \equiv 7 (11) 5

x \equiv 8 (11) 9

x \equiv 9 (11) 4

x \equiv 10 (11) 1

x \equiv 0 (11) 0

x \equiv 5 (11) a n d x \equiv 6 (11)

D (2016, 5) = ⌊ \frac{2016 - 5}{11} ⌋ + ⌊ \frac{2016 - 6}{11} ⌋ = 182 + 182 = 364

Commented by naka3546 last updated on 25/Oct/21

T h a n k y o u s o m u c h, s i r .