Question 550112
Remainders for different numbers divided by different numbers run in patterns.  Start with 3.

4/3 = 1 R 1, 5/3= 1 R2, 6/3 = 2, 7/3=2R1, 8/3 = 2R2.  Starting from 4 every 3rd number has a remainder of 1 when divided by 3:  4, 7, 10, 13....<P>
Find the pattern of remainders when dividing those by 4.<P>
4/4 = R0, 7/4 = 1R3, 10/4=R2, 13/4=R1, 16/4=R0...22/4=R2...every 12 in the 3's remainder 1 series will have a remainder of 2 when divided by 4.  So that's 10,22,34,46,58...<P>
Find the pattern of remainders when dividing those by 5.<P>
10/5=R0, 22/5=R2, 34/5=R4, 46/5=R1, 58/5=R3, 70/5=R0, 82/5=R2, 94/5=R4, 106/5=R1, 118/5=R3...every 60 in the 4's series will have remainder 3 when divided by 5.  So that's 58, 118, 178, 238, 298...<P>
Find the pattern of remainders when dividing those by 6.<P>
58/6=9R4.  That's the smallest number that has remainder 1 when divided by 3, remainder 2 when divided by 4, R3 when divided by 5, and R4 when divided by 6.<P>
58/7=9 R 2
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