Find the largest four-digits number which when divided by 4, 7 and 13 leaves a remainder of 3 in each case.
This question explains how to find the largest four-digit number that leaves a remainder of 3 when divided by 4, 7, and 13. The solution uses the concept of LCM (Least Common Multiple). Since the remainder is the same in each case, subtract 3 from the number to make it exactly divisible by all three numbers. After finding the LCM of 4, 7, and 13 as 364, the greatest multiple of 364 less than 9999 is calculated. Adding 3 back gives the required number as 9831. This problem is useful for understanding divisibility, remainders, and applications of LCM in number system questions.
Question:
Find the largest four-digit number that, when divided by 4, 7, and 13, leaves a remainder of 3 in each case.
Solution:
The largest 4-digit number is 9999
To find the largest 4-digit number divisible by 4, 7 and 13, we find the LCM of 4, 7 and 13 first.
$\operatorname{LCM}(4,7,13)=4 \times 7 \times 13=364$
Now, to we divide 9999 by 364 and subtract the remainder from 9999 to get the number completely divisible by 4, 7 and 13.
$9999-171=9828$
Because the number leaves the remainder 3, we add 3 to 9828.
Therefore, 9828 + 3 = 9831 is the required number.