Read, 4 minutes

Whenever we multiply two whole numbers, we get another number. The numbers we multiplied to get the product are called the factors of the product. For example, \(2 \times 3 = 6\). This implies that 2 and 3 are the factors of 6. Another conclusion we can draw from this is that factors of a number completely divide the number without leaving any remainder.

Example: Let’s consider the number 24. Now, 24 can be divided into factors 6 and 4. Also, 6 can be further factorized into 3 and 2. Moreover, 4 can also be factorized into 2 and 2. So, from this we can see that the other factors of 24 are 3, 8, and 2. This is because \(12 \times 2 = 8 \times 3 = 3 \times 8 = 24\).

Some Facts about Factors:

  • Each and every number has a smallest factor which is 1.
  • Every number has a minimum of two factors, which are 1 and the number itself.
  • Numbers which have only two factors (1 and the number itself) are called prime numbers.

Prime Factorization

Prime factorization is defined as the product of all the prime factors of a number whose multiplication gives the number itself. To write the prime factors of a number, we may have to repeat the number. For example, the factors of 8 are 1 and 2, but to represent 8, we use:

  • \(8 = 2 \times 2 \times 2\)

prime_factors_of_8

Least Common Multiple (LCM)

The LCM is the smallest multiple that two or more numbers have in common. To find the LCM of two numbers, follow these steps:

  1. Find the Greatest Common Factor (GCF) between the two numbers.
  2. Divide either of the two numbers by the GCF.
  3. Multiply the answer by the other number.

Example: Find the LCM of 100 and 50.

  • 100 = 2 × 2 × 5 × 5
  • 50 = 2 × 5 × 5

Thus, the GCF(100, 50) = 2 × 5 × 5 = 50.

Now, 100 ÷ 50 = 2. Therefore, LCM(100, 50) = 50 × 2 = 100.

Exercises for Finding the Least Common Multiple (LCM)

  1. LCM(52, 18) =
  2. LCM(50, 15) =
  3. LCM(81, 30) =
  4. LCM(74, 58) =
  5. LCM(8, 28) =
  6. LCM(15, 35) =
  7. LCM(37, 74) =
  8. LCM(30, 72) =
  9. LCM(23, 46) =
  10. LCM(38, 40) =

Answers

  1. LCM(52, 18) = 468Solution:**Step 1:** Find the GCF of the numbers. GCF: (52, 18) = 2

    **Step 2:** Divide that GCF into either number. \(18 ÷ 2 = 9\)

    **Step 3:** Take that answer and multiply it by the other number. \(9 × 52 = 468\)

  2. LCM(50, 15) = 150Solution:**Step 1:** Find the GCF of the numbers. GCF: (50, 15) = 5

    **Step 2:** Divide that GCF into either number. \(50 ÷ 5 = 10\)

    **Step 3:** Take that answer and multiply it by the other number. \(10 × 15 = 150\)

  3. LCM(81, 30) = 810Solution:**Step 1:** Find the GCF of the numbers. GCF: (81, 30) = 3

    **Step 2:** Divide that GCF into either number. \(30 ÷ 3 = 10\)

    **Step 3:** Take that answer and multiply it by the other number. \(10 × 81 = 810\)

  4. LCM(74, 58) = 2146Solution:**Step 1:** Find the GCF of the numbers. GCF: (74, 58) = 2

    **Step 2:** Divide that GCF into either number. \(58 ÷ 2 = 29\)

    **Step 3:** Take that answer and multiply it by the other number. \(29 × 74 = 2146\)

  5. LCM(8, 28) = 56Solution:**Step 1:** Find the GCF of the numbers. GCF: (8, 28) = 4

    **Step 2:** Divide that GCF into either number. \(28 ÷ 4 = 7\)

    **Step 3:** Take that answer and multiply it by the other number. \(7 × 8 = 56\)

  1. LCM(15, 35) = 105Solution:**Step 1:** Find the GCF of the numbers. GCF: (15, 35) = 5

    **Step 2:** Divide that GCF into either number. \(15 ÷ 5 = 3\)

    **Step 3:** Take that answer and multiply it by the other number. \(3 × 35 = 105\)

  2. LCM(37, 74) = 74Solution:**Step 1:** Find the GCF of the numbers. GCF: (37, 74) = 37

    **Step 2:** Divide that GCF into either number. \(74 ÷ 37 = 2\)

    **Step 3:** Take that answer and multiply it by the other number. \(2 × 37 = 74\)

  3. LCM(30, 72) = 360Solution:**Step 1:** Find the GCF of the numbers. GCF: (30, 72) = 6

    **Step 2:** Divide that GCF into either number. \(72 ÷ 6 = 12\)

    **Step 3:** Take that answer and multiply it by the other number. \(12 × 30 = 360\)

  4. LCM(23, 46) = 46Solution:**Step 1:** Find the GCF of the numbers. GCF: (23, 46) = 23

    **Step 2:** Divide that GCF into either number. \(46 ÷ 23 = 2\)

    **Step 3:** Take that answer and multiply it by the other number. \(2 × 23 = 46\)

  5. LCM(38, 40) = 760Solution:**Step 1:** Find the GCF of the numbers. GCF: (38, 40) = 2

    **Step 2:** Divide that GCF into either number. \(40 ÷ 2 = 20\)

    **Step 3:** Take that answer and multiply it by the other number. \(20 × 38 = 760\)