What Is The Least Common Multiple Of 7 And 9?

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Introduction

In mathematics, a multiple is the product of any two numbers. For example, the multiples of 7 are 7, 14, 21, 28, and so on. Similarly, the multiples of 9 are 9, 18, 27, 36, and so on. However, sometimes we need to find the least common multiple of two numbers, which is the smallest multiple that is common to both numbers. In this article, we will discuss how to find the least common multiple of 7 and 9.

Divisibility Rules

To find the least common multiple of 7 and 9, we need to understand the concept of divisibility. A number is divisible by another number if it can be divided evenly without leaving a remainder. For example, 9 is divisible by 3 because 9 ÷ 3 = 3 with no remainder. Similarly, 7 is not divisible by 3 because 7 ÷ 3 = 2 with a remainder of 1. There are certain rules for divisibility that can help us determine whether a number is divisible by another number.

Divisibility Rule for 7

The divisibility rule for 7 states that if you take the last digit of a number, double it, and subtract it from the remaining digits, the result should be divisible by 7. For example, let's take the number 35. The last digit is 5, so we double it to get 10. Then we subtract 10 from 35 to get 25. Since 25 is not divisible by 7, we know that 35 is not divisible by 7.

Divisibility Rule for 9

The divisibility rule for 9 states that if you add up all the digits in a number and the sum is divisible by 9, then the number itself is divisible by 9. For example, let's take the number 54. The sum of its digits is 5 + 4 = 9, which is divisible by 9. Therefore, we know that 54 is divisible by 9.

Finding the Least Common Multiple

To find the least common multiple of 7 and 9, we can use a method called prime factorization. Prime factorization is the process of breaking down a number into its prime factors, which are the smallest prime numbers that can be multiplied together to get the original number. For example, the prime factors of 12 are 2, 2, and 3 because 2 × 2 × 3 = 12.

Prime Factors of 7

The number 7 is a prime number, which means it can only be divided evenly by 1 and itself. Therefore, the prime factorization of 7 is simply 7.

Prime Factors of 9

The number 9 is not a prime number because it can be divided evenly by 3. Therefore, we can break down 9 into its prime factors by dividing it by 3. We get: 9 ÷ 3 = 3. Since 3 is a prime number, we have found the prime factorization of 9, which is 3 × 3.

Finding the Least Common Multiple

To find the least common multiple of 7 and 9, we need to multiply together all the prime factors of both numbers, but we only need to include each prime factor once. Therefore, the least common multiple of 7 and 9 is 7 × 3 × 3 = 63.

Conclusion

In conclusion, the least common multiple of 7 and 9 is 63. To find the least common multiple, we used the concepts of divisibility and prime factorization. We also learned the divisibility rules for 7 and 9, which can be helpful in determining whether a number is divisible by these numbers. By understanding these concepts, we can find the least common multiple of any two numbers.