Help to solving an important equation for whole numbers...?

Question

Hi there. I have equation which I want to solve it for whole numbers, See that:

k = t / (p-t)

in this equation k,t and p are whole numbers. and p is constant (Not Pi in trig) and the variable to solve is t, in other word this equation wants that the whole numbers named t, which t/(p-t) be a whole number.

Be succeed,
Babax.

Answer 1

To solve for t:
t / (p-t) = k
<=> t = k(p-t) = kp - kt
<=> (k+1) t = kp
<=> t = kp / (k+1)
Since (k+1) and k are coprime, k+1 must be a factor of p for there to be any solutions.

(In the case k = -1, where this won't work, we get t = -(p-t) = t - p, so any t will work if p = 0; otherwise there are no solutions.)

In general, if you're looking for solutions with any values of k, p and t, note that k must be a factor of t and (k+1) must be a factor of p. So if you're given p but not t or k, for instance, start by taking the factors of p and subtracting 1 to get possible values of k. Then t = kp / (k+1) as before.

Answer 2

dunno but the first 2 posters seem to get their algebra from some chinese school

k=t/(p-t)
multiple both sides with (p-t)
k(p-t)=t
kp-kt=t
you can have an additional step here where kp=t+kt but why bother?
kt+t=kp
t(k+1)=kp
t=kp/(k+1)

same working, but numeric value is always on the right of the variable.

also, never leave your answer inverted, ie kp/(k+1)=t

Answer 3

k = t/(p-t)

Multiply (p-t) to both sides

k(p-t) = t

Multiply the left side out

kp - kt = t

Add kt to both sides

kp = t + kt

Factor the left side by t

kp = t(1+k)

Divide both sides by (1+k)

kp / (1+k) = t

There you go!

Answer 4

I don't know. ther is an element of surprise her because it cannot be correct to leave k over k in the final answer.

Answer 5

k = t / (p-t)

k(p-t) = t

kp - kt = t

kp = t(1+k)

t = kp/(1+k)