A 4-digit number N is such that when divided by 3, 5, 6, 9 leaves a remainder 1, 3, 4, 7 respectively. What…
A 4-digit number N is such that when divided by 3, 5, 6, 9 leaves a remainder 1, 3, 4, 7 respectively. What is the smallest value of N?
1068
1072
1078
1082
Solution
In each case the remainder is exactly 2 less than the divisor ($3-1=2$, $5-3=2$, $6-4=2$, $9-7=2$). So $N+2$ is divisible by 3, 5, 6 and 9. LCM(3,5,6,9) = 90, so $N+2$ is a multiple of 90, i.e. $N = 90k - 2$. The smallest 4-digit value: $90\times12 - 2 = 1080 - 2 = 1078$.