find the greatest four digit number which when divided by 11 and 15 leaves reminder 2 and 4 respectively and also the number exactly divisible by 13

Question

find the greatest four digit number which when divided by 11 and 15 leaves reminder 2 and 4 respectively and also the number exactly divisible by 13.

Answer 1

start with 9999, but this is not divisible by 13

9997 is divisible by 13, 9997/13=769

first find the number that satisfies condition one, that when divided by 11 leaves a remainder of 2

subtract 13 until you get to 9594 which satisfies the first condition

from this number every 11 subtractions of 13, or 143, satisfies condition one, but not condition two

9544, 9451, 9308, 9165, 9022, 8879, 8736, 8593, 8550, and 8307 all satisfy condition one but not condition two and all of these numbers are divisible by 13 because we started with 9997 and subtracted 13's which always gives a number divisible by 13

the next subtraction of 143 gives 8164 which is divisible by 13 and when divided by 11 leaves a remainder of 2 and when divided by 15 leaves a remainder of 4

8164/13=628

8164/11=742 remainder 2

8164/15=544 remainder 4

in doing the problem on a calculator, a remainder of 2 when divided by 11 gives 0.181818... and a remainder of 4 when divided by 15 gives 0.2666..., so you look for these decimal parts when you divide to make your work easier