Divisibility

Number Theory Level 2

For how many natural numbers x x , is the expression: x 2 + 2 x + 3 x ^2 + 2x + 3 divisible by 35 35 ?

0 7 1 35

Danish Ahmed by Danish Ahmed , 24, India

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1 solution

Danish Ahmed
Jan 21, 2015

The number x x can take any value

So, x 2 x^2 will always end in: 0 0 , 1 1 , 4 4 , 5 5 , 6 6 , or 9 9 .

So, x 2 + 2 x x^2 + 2x will end in: 0 0 , 3 3 , 4 4 , 5 5 ,or 8 8 .

Therefore, x 2 + 2 x + 3 x^2 + 2x + 3 will always end in: 1 1 , 2 2 , 3 3 , 6 6 , 7 7 , or 8 8 .

Since in none of the cases the last digit is a 0 0 or a 5 5 , hence it is not divisible by 5 5 and therefore never by 35 35 implying that there does not exist any natural number x x such that x 2 + 2 x + 3 x^2 + 2x + 3 will be divisible by 35 35 without leaving any remainder.

4 Helpful 0 Interesting 0 Brilliant 0 Confused

x 2 + 2 x + 3 ≡ 0 ( m o d ⁣ 35 ) ⇒ ( x + 1 ) 2 ≡ 3 ( m o d 5 ) x^2+2x+3\equiv 0\pmod{\! 35}\,\Rightarrow\, (x+1)^2\equiv 3\pmod{5} , but ( 3 / 5 ) = − 1 (3/5)=-1 (Legendre symbol), so no solutions can exist.

mathh mathh - 6 years ago

I did it in exactly ur same way. GG

Pranav Jayaprakasan UT - 4 years, 3 months ago

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