Do you want to know how to find the greatest 5-digit number that, when divided by 3, 7, 8, and 12, leaves a remainder of 2? If so, you've come to the right place! This article will walk you through the process step-by-step and explain the math behind it.

Introduction

The problem we're addressing is finding the largest 5-digit number (which is 99999) such that when divided by 3, 7, 8, and 12, the remainder is always 2. This requires a bit of number theory and a good understanding of the Least Common Multiple (LCM).

Step 1: Find the LCM

First, we need to find the Least Common Multiple (LCM) of the numbers 3, 7, 8, and 12. The LCM is the smallest number that is a multiple of all the given numbers. Here's how to calculate it:

LCM(3, 7, 8, 12) 168

How did we get 168? We need to find the smallest number that is divisible by 3, 7, 8, and 12. Let's break it down:

3 3 7 7 8 2^3 12 2^2 * 3

So, the LCM must include the highest powers of all prime factors:

2^3 (from 8) 3 (from 3 and 12) 7 (from 7)

The LCM is 2^3 * 3 * 7 168.

Step 2: Subtract the Quotient

Now that we have the LCM, we can use it to find the remainder. We start by dividing the largest 5-digit number (99999) by the LCM (168) and look at the remainder:

99999 ÷ 168 597 R 39 (Quotient 597, Remainder 39)

This means that the largest number divisible by 168 is:

168 * 597 99960

Now, we need a number that leaves a remainder of 2 when divided by 168. To get this, we subtract the remainder (39) from 99999 and then add 2:

99999 - 39 2 99962

Conclusion

By following these steps, we found that the greatest 5-digit number that, when divided by 3, 7, 8, and 12, leaves a remainder of 2 is 99962.

Additional Examples

Let's look at a few more examples:

Example 1: Given numbers are 3, 7, 8, 12; LCM 168; Dividend 99999; Quotient 597; Remainder 39; Result 99999 - 39 2 99962

Additional Tips

When working with larger numbers, always use a calculator or software to avoid errors. Also, remember to systematically subtract the remainder from the largest number and then add the required remainder.

Related Topics

Number Theory Divisibility Rules Remainders and Divisors

By understanding the LCM and how it relates to remainders, you can solve similar problems more efficiently. If you have any questions or need further assistance, feel free to ask!