Question

find the greatest number less than 1000 which divided by 5 8 9 12 15 leaves a reminder 4 in each case

Answer 1

Answer:

We see that the remainders are 22 less than the number by which we are dividing. So it is a special case, where the required number is 22 less than a common multiple of the given factors, viz. 55, 77, and 99.

The least common multiple of 5, 7, and 9 is 5×7×9=3155×7×9=315.

Now, 1000315=311631000315=31163

Hence the required common multiple must be 3×315=9453×315=945.

So the required number is 22 less than 945945, i.e. 943943.

Verification:

943=5×188+3943=5×188+3

943=7×134+5943=7×134+5

943=9×104+7