Thread: integers

When a certain postive integer N is divided by a postive integer d, the remainder is 7. If 2N + 3 is divided by d, the remainder is 1. Find all possible values of d.

I don't know in what branch of Math is this problem, so I will solve it the way I know how to solve it. I will solve it my way.
If my solution is not what you expect, then just consider my solution as another way to solve your problem here.

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"When a certain postive integer N is divided by a postive integer d, the remainder is 7."

N/d = a +7/d ...(i)
where
"a" is another positive integer.

Multiply both sides of (i) by d,
N = a*d +7 ....(ii)

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"If 2N + 3 is divided by d, the remainder is 1."

(2N +3)/d = b +1/d ....(iii)
where
"b" is yet another positive integer.

Multiply both sides of (iii) by d,
2N +3 = b*d +1 ....(iv)

Substitute the N from (ii) into (iv),
2(a*d +7) +3 = b*d +1
2a*d +14 +3 = b*d +1
Collect the d-terms,
2a*d -b*d = 1 -14 -3
d(2a -b) = -16
d = -16/(2a -b)
Or,
d = 16/(b -2a) ....(v)

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d = 16/(b -2a) ....(v)

Examine the denominator (b -2a).

>>>Since "a" and "b" are positive integers, then (b -2a) can never be a fraction. (b -2a) is another integer.

>>>Since "d" is a positive integer, then (b -2a) can never be negative. (b -2a) is always positive.

>>>Since "d" is an integer, then 16/(b -2a) is an integer. 16/(b -2a) cannot be a fraction.

>>>Since 16/(b -2a) is an integer, then (b -2a) can never be more than 16.
So, maximum (b -2a) is 16.

>>>Since 16/(b -2a) is an integer, then (b -2a) is an integer factor of 16.

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Now we can find all the possible values of "d".

d = 16/(b -2a) ....(v)

If (b -2a) = 16,
d = 16/16 = 1 ....***

If (b -2a) = 15,
d = 16/15 = not integer.

If (b -2a) = 14, 13, 12, 11, 10, 9,
d = 16/14, 16/13, 16/12, 16/11, 16/10, 16/9
All are not integers.

If (b -2a) = 8,
d = 16/8 = 2 ....***

If (b -2a) = 7, 6, 5,
d = 16/7, 16/6, 16/5 = not integers.

If (b -2a) = 4,
d = 16/4 = 4 ....***

If (b -2a) = 3,
d = 16/3 = not integer.

If (b -2a) = 2,
d = 16/2 = 8 ....***

If (b -2a) = 1,
d = 16/1 = 16 ....***

Therefore, "d" = 1, 2, 4, 8, or 16. ....answer.

Thanks for the help, I understand it now

There's a shorter way of doing this. We're told d divides N - 7 and (2N+3) - 1 which is 2N+2. So d divides (2N+2) - 2.(N-7) = 16.