Question

Show that \(2450^n-1370^n-1150^n-250^n\) is divisible by 1980 for all \(n\epsilon N\)

Answer 1

Are you trying to denote the set membership of n in N? If so use \in, also what is N?

Answer 2

n belongs to the set of Natural numbers

Answer 3

You should use \mathbb{N}

Answer 4

$$n\in \mathbb{N}$$

Answer 5

Do you know some modular arithmetic?

Answer 6

yeah.. I just don't know how to choose the correct figures ..

Answer 7

Since \(1980=4\cdot 9\cdot 5\cdot 11\), one way to do this is to try and show that the statement is divisible by 4, 9, 5 and 11.

Answer 8

oops its 1980 my bad

Answer 9

Reduce 2450 modulo 1980, then if you can show the expression is zero modulo 4,9,5 and 11 your done.

Answer 10

can't i directly show that the statement is divisible by 5 and 396 and hence prove that it is also divisible by 1980?

Answer 11

That works too.

Answer 12

yes, sense there both coprime

Answer 13

Thanks! So, if i come across such problems, all i need to do is find the factors first and then prove that the expression is divisible by a prime factor among them and the product of the rest all factors.. (as long as the product of rest all factors is not a multiple of the prime number itself)

Answer 14

Yes, but for a proof of this and other techniques I would read up on some elementary number theory, if you plan on using it in the future or somthing.

Answer 15

elementary number theory like?

Answer 16

Modular arithmetic

Answer 17

yeah, i know modular arithmetic.. but i didn't know how to apply that in solving this.. Now i do!

Answer 18

If gcd(a,b)=1, and a divides n, and b divides n. Then ab divides n, this is essentially what were using here, to simplify the problem into cases where the expression is divisible by smaller integers.

Answer 19

Ah, yes!! Thanks!! OS is always helpful! :)

Answer 20

I might be wrong, but I dont think your statement is divisible by 9. Maybe one of those minus signs should be a plus? Then it will work nicely.

Answer 21

If the problem was
\[2450^n-1370^n+1150^n-250^n\]it would make more sense.

Answer 22

Ah yes, That was a typo.. Sorry! Its \(+1150^n\)