What is a nonexample of a least common multiple?

Answer 1

The least common multiple of two or more nonzero whole numbers is the smallest whole number that is divisible by each of the numbers. There are two common methods for finding the least common multiple of 2 numbers.

Method 1:

List the multiples of each number, and look for the smallest number that appears in each list.

Example:

Find the least common multiple of 12 and 42. We list the multiples of each number:

12: 12, 24, 36, 48, 60, 72, 84, ...

42: 42, 84, 126, 168, 190, ...

We see that the number 84 is the smallest number that appears in each list.


Method 2:

Factor each of the numbers into primes. For each different Prime number in either of the factorizations, follow these steps:

1. Count the number of times it appears in each of the factorizations.

2. Take the largest of these two counts.

3. Write down that prime number as many times as the count in step 2.

To find the least common multiple take the product of all of the prime numbers written down in steps 1, 2, and 3.


Example:

Find the least common multiple of 24 and 90. First, we find the prime factorization of each number.

24 = 2 × 2 × 2 × 3

90 = 2 × 3 × 3 × 5

The prime numbers 2, 3, and 5 appear in the factorizations. We follow steps 1 through 3 for each of these primes.

The number 2 occurs 3 times in the first factorization and 1 time in the second, so we will use three 2's.

The number 3 occurs 1 time in the first factorization and 2 times in the second, so we will use two 3's.

The number 5 occurs 0 times in the first factorization and 1 time in the second factorization, so we will use one 5.

The least common multiple is the product of three 2's, two 3's, and one 5.

2 × 2 × 2 × 3 × 3 × 5 = 360

Example:

Find the least common multiple of 14 and 49. First, we find the prime factorization of each number.

14 = 2 × 7

49 = 7 × 7

The prime numbers 2 and 7 appear in the factorizations. We follow steps 1 through 3 for each of these primes.

The number 2 occurs 1 times in the first factorization and 0 times in the second, so we will use one 2.

The number 7 occurs 1 time in the first factorization and 2 times in the second, so we will use two 7's.

The least common multiple is the product of one 2 and two 7's.

2 × 7 × 7 = 98

Examples:

Some other least common multiples are listed below.

The least common multiple of 12 and 9 is 36.

The least common multiple of 6 and 18 is 18.

The least common multiple of 2, 3, 4, and 5 is 60.

Answer 2

450 is not the LCM of 17 and 23.

689 is not the LCM of 15 and 20.

This is much easier than the other way.

Answer 3

Non-integers do not have a least common multiple.

Answer 4

Any set of non-zero integers has an LCM.

Answer 5

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Answer 6

Predict