Find the least positive integer that leaves remainders of 1, 2 and 3 when divided by 3, 5 and 7 respectively.

Answer 1

With relatively small numbers like in your problem, we could just "grind out" the answer in the following way:

numbers which leave a remainder of 1
when divided by 3:
4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 ....
numbers which leave a remainder of 2 when divided by 5 :
7 12 17 22 27 32 37 42 47 52 57 ..
numbers which leave a remainder of 3 when divided by 7:
10 17 24 31 38 45 52 59 ...

notice that 52 is the first first, thus the smallest number that satisfies your conditions.

If the numbers had been larger, the above method would not be practical and we would use other methods dealing with modular arithmetic. If interested look up the "Chinese Remainder Algorithm", makes for some real math-fun.