# maths

posted by .

A positive integer n is such that numbers 2n+1 and 3n+1 are perfect squares. Prove that n is divisible by 8.
thanks

• maths -

I sort of "cheated" by running a quick computer program.
in the first 1 million values of n
I found only 3 values of n that satisfied your condition.
they were 40, 3960, and 388080

All 3 are divisible by 8

I can't think of a "mathematical" way to do you question.

• maths -

Modulo 8, a square can only be 0, 1 or 4:

0^2 = 0

1^2 = 1

2^2 = 4

3^2 = 1

4^2 = 0

The squares of 5, 6 and 7 are the same as the squares of 3, 2 and 1.

Since 2n + 1 is odd and is a square, it must be 1 modulo 8. So, Mod 8 we have:

2 n + 1 = 1 ------>

2 n = 0

So, we know that n is a multiple of 4, so we can be sure that n is even. But this means that 3n + 1 must be odd. Because 3 n + 1 is a square, it follows that 3 n + 1 Modulo 8 equals 1. So, Modulo 8 we have:

2 n + 1 = 1

3 n + 1 = 1

Subtracting gives (Modulo 8):

n = 0

So, n is a multiple of 8.

## Similar Questions

1. ### number theory

How might you go about proving that N-M is divisible by 9 when N represents an integer (such as 6923) and M represents that integer in reverse order (such as 3296)?
2. ### math

can you answer this question in a different and more logical way than this method below: we will examine the sum of cubes of two numbers, A and B. Without losing generality, we will further assume that A=2nX and B=2n+kY where X is …
3. ### maths

can you answer this question: prove that a number 10^(3n+1), where n is a positive integer, cannot be represented as the sum of two cubes of positive integers. with out using this method at all ................................. We …
4. ### math

According to the Journal of Irreproducible Results, any obtuse angle is a right angle! Here is their argument. Given the obtuse angle x, we make a quadrilateral ABCD with  DAB = x, and  ABC = 90◦, andAD = BC. Say the perpendicular …
5. ### math

Find a positive integer m such that 1 2m is a perfect square and 1 3m is a perfect cube. Can you find a positive integer n for which 1 2n is a perfect square, 1 3n is a perfect cube and 1 5n is a perfect fifth power?
6. ### MATH

Find a positive integer m such that 1/2m is a perfect square and 1/3m is a perfect cube. Can you find a positive integer n for which 1/2n is a perfect square, 1/3n is a perfect cube and 1/5n is a perfect fifth power?
7. ### maths-

if n is an odd positive integer, then prove that n^2 -1 is divisible by 8
8. ### maths

What is the sum of all possible positive integer values of n, such that n^2+19n+130 is a perfect square?
9. ### Math

Let z be a complex number, and let n be a positive integer such that z^n = (z + 1)^n = 1. Prove that n is divisible by 6. I have no idea how to approach this problem!
10. ### math

2x is a perfect square, 3x a perfect cube, and 5x a perfect 5th power. Find the sum of the exponents in the prime factorization of the smallest such positive integer x. Thank you.

More Similar Questions