Introduction to Finding the Least Number with a Specific Remainder

Understanding how to find the least number that leaves a specific remainder when divided by given divisors is a crucial skill in mathematics, particularly helpful in solving problems related to least common multiples (LCMs) and modular arithmetic. This guide will walk you through the process using a real-life example.

Understanding the Problem

The problem we will solve is: What is the least number which when divided by 9, 12, 16, and 20, leaves 3 as the remainder in each case?

Step-by-Step Solution

To solve this problem, we will follow several steps:

Identify the given divisors and the remainder. Calculate the least common multiple (LCM) of the divisors. Subtract the given remainder from the LCM to find the number that meets the condition.

Step 1: Divisors and Remainder

The divisors are 9, 12, 16, and 20. The remainder we are interested in is 3.

Step 2: Calculating the LCM

The LCM is the smallest number that is evenly divisible by the given divisors. To find the LCM, we first need to determine the prime factorization of each divisor:

9 32 12 22 × 3 16 24 20 22 × 5

Next, we find the LCM by taking the highest power of each prime that appears in the factorizations:

For 2: the highest power is 24 from 16. For 3: the highest power is 32 from 9. For 5: the highest power is 51 from 20.

Therefore, the LCM is:

LCM 24 × 32 × 51 16 × 9 × 5

Calculating the LCM step-by-step:

16 × 9 144 144 × 5 720

Hence, the LCM of 9, 12, 16, and 20 is 720.

Step 3: Finding the Desired Number

We now know that the number we are looking for, when divided by 9, 12, 16, or 20, will leave a remainder of 3. We use the formula:

x LCM remainder

Substituting the values:

x 720 3 723

Thus, the least number which when divided by 9, 12, 16, and 20, leaves a remainder of 3 in each case is:

723

Verifying the Solution

Let's verify that 723 has a remainder of 3 when divided by 9, 12, 16, and 20:

723 ÷ 9 80 remainder 3 723 ÷ 12 60 remainder 3 723 ÷ 16 45 remainder 3 723 ÷ 20 36 remainder 3

As we can see, 723 satisfies the conditions.

Conclusion

To find the least number with a given remainder when divided by several composite numbers, we use the LCM method and then adjust for the remainder. This technique is useful in various mathematical and real-world applications.