# Thread: Number of Integers

1. ## Number of Integers

How many integers n is 7n+1 divisible by 3n+5?

What exactly is this question asking?
What's the set up?

2. ## Re: Number of Integers Originally Posted by harpazo How many integers n is 7n+1 divisible by 3n+5?

What exactly is this question asking?
What's the set up?
Hint: Do the long division $\displaystyle \dfrac{7n + 1}{3n + 5}$. You'll get a remainder that depends on n. How many values of n are there such that the remainder is 0?

-Dan

Addendum: Or you could just pick a bunch of n values and see how many work.

3. ## Re: Number of Integers Originally Posted by harpazo How many integers n is 7n+1 divisible by 3n+5?
(7n - 1)/(3n + 5) = x
solve for n:
n = (5x - 1)/(7 - 3x)

So (from denominator) 3x < 7 : see that??

4. ## Re: Number of Integers Originally Posted by DenisB (7n - 1)/(3n + 5) = x
solve for n:
n = (5x - 1)/(7 - 3x)

So (from denominator) 3x < 7 : see that??
n is not restricted to positive integers. If n = -2, for instance, then x = 13.

5. ## Re: Number of Integers

Yep....so:
n = -7 or -3 or -2 or -1 or 1 or 2

6. ## Re: Number of Integers Originally Posted by DenisB Yep....so:
n = -7 or -3 or -2 or -1 or 1 or 2
You have made two errors. First, you included n = 2, which does not work
because the fraction equals 15/11, a noninteger. Second, you did not include the
solution of n = 9, which makes the fraction 64/32 = 2.

Also, I did not catch your typo in the numerator of your fraction.
Instead of (7n - 1) that you wrote, it is supposed to be (7n + 1).
However, the expression for n in terms of x right below that is correct.

7. ## Re: Number of Integers

Thank you everyone.

8. ## Re: Number of Integers

Yikes...my "2" should be "9" (typo).

-7 -48/-16 3
-3 -20/-4 5
-2 -13/-1 13
-1 -6/2 -3
1 8/8 1
9 64/32 2

Hope that's typoless!

Thanks Greg.

Good notes.