+489 Determine the smallest non-negative integer that satisfies the congruences:
\(\begin{align*} &a\equiv 2\pmod 3,\\ &a\equiv 4\pmod 5,\\ &a\equiv 6\pmod 7,\\ &a\equiv 8\pmod 9. \end{align*}\)
Since the LCM of 3, 5, 7, 9 =315, and the remainders are 1 less than the moduli in each case, therefore the smallest number that satisfies the congruences is:
315 - 1 = 314, so that you have:
315N + 314 and N =0, 1, 2, 3......etc. So:
a = 314, 629, 944.....etc.
+118609 Thanks Guest, I just wanted to think this one through myself :)
a=9k1-1 that satisifes both the first and the last congruences
a=5k2-1
a=7k3-1
So if a is 1 less than a multiple of 5 and 9 and 7 then it must be 1 less than any common factor of 9 and 5 and 7
So the smallest one is 1 less than 315
answer 314
