Let a and b be integers If d sa tb where st are integers f

Let a and b be integers. If d = sa + tb, where s,t are integers, find infinitely many pairs of integers (s_k, t_k) with d = (s_k)a + (t_k)b

Solution

if d = sa + tb we have to find the infinitely many pairs of (s_k , t_k) with d = (s_k)a + (t_k)b

we have to find s_k and t_k such that (s_k)a + (t_k)b = d = sa + tb

we can assume that,

s_k = s + kb

t_k = t - ka

so,

(s_k)a + (t_k)b = (s + kb)a + (t - ka)b

(s_k)a + (t_k)b = sa + kba + bt - kba

(s_k)a + (t_k)b = sa + bt

as given we have sa + bt = d so,

(s_k)a + (t_k)b = d

so we can say that,

(s_k , t_k) = (s+kb , t-ka)