1. ## Value of n

When 10 is divided by the positive integer n, the remainder is (n - 4). What could be the value of n?

10/n

Remainder = n - 4

Is the equation set up (10/n) = n - 4?

2. ## Re: Value of n

Originally Posted by harpazo
When 10 is divided by the positive integer n, the remainder is (n - 4). What could be the value of n?

10/n

Remainder = n - 4

Is the equation set up (10/n) = n - 4?
No, the dividend divided by a divisor equals a quotient, plus any remainder.

Or, put another way, the divisor multiplied by the quotient, plus the remainder, equal the dividend.

Here, 10 is the dividend, n is the divisor, the remainder is (n - 4), but the quotient is not explicitly given.

I'll call the quotient Q.

n*Q + (n - 4) = 10

n*Q + n = 14

n(Q + 1) = 14

$\displaystyle n \ = \ \dfrac{14}{Q + 1}$

3. ## Re: Value of n

Originally Posted by greg1313
No, the dividend divided by a divisor equals a quotient, plus any remainder.

Or, put another way, the divisor multiplied by the quotient, plus the remainder, equal the dividend.

Here, 10 is the dividend, n is the divisor, the remainder is (n - 4), but the quotient is not explicitly given.

I'll call the quotient Q.

n*Q + (n - 4) = 10

n*Q + n = 14

n(Q + 1) = 14

$\displaystyle n \ = \ \dfrac{14}{Q + 1}$

To find n, I need to know Q. How do I find Q?

4. ## Re: Value of n

Originally Posted by harpazo
To find n, I need to know Q. How do I find Q?
Suppose you start out with the interval $\displaystyle \ 1 \le n \le 10$.

For usual division, the remainder is non-negative, so $\displaystyle \ (n - 4) \ge 0$.

Or, $\displaystyle \ \ n \ge 4$.

If 10 is divided by an n in this interval, the quotient would be in this interval: $\displaystyle \ 1 \le Q \le 10$.

(Q + 1) must be positive and divide 14. So, candidates for Q are 1 and 6. But, a Q-value of 6 makes an n-value of 2,
and we already stated that $\displaystyle \ \ n \ge 4$.

Check out what happens if Q = 1.

5. ## Re: Value of n

Originally Posted by greg1313
Suppose you start out with the interval $\displaystyle \ 1 \le n \le 10$.

For usual division, the remainder is non-negative, so $\displaystyle \ (n - 4) \ge 0$.

Or, $\displaystyle \ \ n \ge 4$.

If 10 is divided by an n in this interval, the quotient would be in this interval: $\displaystyle \ 1 \le Q \le 10$.

(Q + 1) must be positive and divide 14. So, candidates for Q are 1 and 6. But, a Q-value of 6 makes an n-value of 2,
and we already stated that $\displaystyle \ \ n \ge 4$.

Check out what happens if Q = 1.
If Q is 1, then n is 7. So, the answer is 7.

6. ## Re: Value of n

Originally Posted by harpazo
If Q is 1, then n is 7. So, the answer is 7.
Yes. Quick check: When you divide 10 by 7, you get a remainder of 3, which is 7-4.

7. ## Re: Value of n

Originally Posted by Debsta
Yes. Quick check: When you divide 10 by 7, you get a remainder of 3, which is 7-4.
Thank you for your help and patience.

8. ## Re: Value of n

how about $n=14$ ?

9. ## Re: Value of n

Originally Posted by Idea
how about $n=14$ ?
An interesting thought....

-Dan

10. ## Re: Value of n

Originally Posted by Idea
how about $n=14$ ?
Are you saying to let n = 14 for n = 14/(Q + 1)?

n = 14/(Q + 1)

14 = 14/(Q + 1)

14(Q+ 1) = 14

14Q + 14 = 14

14Q = 14 - 14

14Q = 0

Q = 0/14

Q = 0

When Q = 0, then we have:

n = 14/(Q + 1)

n = 14/(0 + 1)

n = 14/1

n = 14

11. ## Re: Value of n

Originally Posted by harpazo
Are you saying to let n = 14 for n = 14/(Q + 1)?

n = 14/(Q + 1)

14 = 14/(Q + 1)

14(Q+ 1) = 14

14Q + 14 = 14

14Q = 14 - 14

14Q = 0

Q = 0/14

Q = 0

When Q = 0, then we have:

n = 14/(Q + 1)

n = 14/(0 + 1)

n = 14/1

n = 14
yes that's how we check and see that n=14 is a solution

10 divided by n=14 gives a quotient Q=0 and remainder = 10 = n - 4

the same way that

10 divided by n=7 gives Q=1 and remainder = 3 = n - 4

12. ## Re: Value of n

Originally Posted by Idea
yes that's how we check and see that n=14 is a solution

10 divided by n=14 gives a quotient Q=0 and remainder = 10 = n - 4

the same way that

10 divided by n=7 gives Q=1 and remainder = 3 = n - 4
Very good. Interesting reply. Interesting question.