Factors and Multiples

**Category : **5th Class

**Factors and Multiples**

**Factors:** The numbers that are multiplied to give a product are called factors.

e.g.,

Factors divide the number exactly (i.e., without leaving a remainder.) So, factors are also called divisors.

1 is a factor of every number, and every number is a factor of itself.

1 is the smallest factor of a number and the number itself is its greatest factor.

The factor of a number is less than or equal to the number.

Every number (except 1) has at least 2 factors - 1 and the number itself.

The factors or divisors that are common to two or more numbers are called their common factors.

∴ Common factors of 4 and 6 are 1, 2.

**Highest Common Factor (H.C.F):**The highest of the common factors of two or more numbers is called their Highest Common Factor (H.C.F.) or their Greatest Common Divisor (G.C.D.).

**Multiples:**The products obtained when a number is multiplied by 1, 2, 3, 4 and so on are called the multiples of that number.

**e.g.,** 4, 8, 12, 16, 20,.... are the multiples of 4.

8, 16, 24, 32, 40,... are the multiples of 8.

Every number is a multiple of 1.

A number is the smallest multiple of itself.

Every multiple of a number is greater than or equal to the number itself.

Multiples of a number are infinite. There is no largest multiple of a number.

The multiples that are common to two or more numbers are called their common multiples.

**Least Common Multiple (L.C.M.):**The lowest of the common multiples of two or more numbers is called their Lowest (or Least) Common Multiple (L.C.M.).

**Even numbers:**The numbers which are multiples of 2 are called even numbers.

**e.g.,** 438, 1450, 7034 etc.

**Odd numbers:**The numbers other than the multiples of 2 are called odd numbers.

**e.g.,** 215, 6013, 897 etc.

**Prime numbers:**The numbers which have only 1 and itself as factors are called prime numbers.

**e.g.,** 3, 11, 23, 47, etc.

**Composite numbers:**The numbers which have at least 1 factor other than 1 and itself are called composite numbers.

**e.g.,** 4, 9, 76,108 etc.

(a) 1 is neither prime nor composite.

(b) 2 is the smallest and the only even prime number.

(c) 4 is the smallest composite number.

(d) Prime numbers other than 2 are odd.

**Twin primes:**Two consecutive prime numbers that differ by 2 are called twin primes.

**e.g.**, (3, 5), (5, 7), (11, 13) etc.

**Co-prime numbers:**The numbers which have no common factor except 1 are called co-prime numbers.

e.g., (4, 15),(11,17),(18,37)etc.,

**Prime factorization:**The process of splitting a given number into its prime factors is called prime factorization.

**Methods to find the H.C.F. of the given numbers:**

**(a) Listing the factors:**

**e.g.,** Find the H.C.F of 36 and 72.

**Step 1:** **List all the factors of the given numbers.**

**Factors of 36: **1, 2, 3, 4, 6, 9, 12, 18, 36

**Factors of 72:** 1, 2, 3, 4, 6, 9, 12, 18, 36, 72

**Step 2:** **Find the common factors of the given numbers.**

The common factors of 36 and 72 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

**Step 3:** **The greatest of the common factors is the required H.C.F.**

36 is the greatest of the common factors of 36 and 72.

∴ 36 is the required H. C. F.

**(b) Prime factorization:**

e.g., find the H. C. F. of 36 and 72.

**Step 1:** **Express the given numbers as the product of prime numbers.**

36 =2×2×3×3

72 =2××2×3×3

**Step 2:** **Circle their common prime factors.**

**Step 3:** **Consider one set of the common factors and find their product.**

2×2×3×3=36

**Step 4:** **The product obtained in step 3 is the required H. C. F.**

Hence, the H. C. F. of 36 and 72 is 36.

**(c)Division method:**

**e.g..** Find the H.C.F of 36, 72 and 144.

**Step 1: Divide the larger number by the smallest number.**

**Step 2:** If the remainder is not zero, divide the divisor by the remainder. Continue the procedure until the remainder obtained is 0.

If the remainder obtained is 0, divide the other given number by the divisor.

**Step 3:** If the remainder is 0, the divisor is the Highest Common Factor or Greatest Common Divisor of the given numbers.

Thus, 36 is the H. C. F. (or G. C. D.) of 36, 72 and 144.

**Methods to find the L.C.M. of the given numbers:**

**(a) Listing the multiples:**

**e.g.,** find the L.C.M. of 36 and 72.

**Step 1:** List the first few multiples of the given numbers.

**Multiples of 36:** 36, 72,108,144,180, 216,...

**Multiples of 72:** 72,144, 216,...

**Step 2:** Find the common multiples of the given numbers.

The common multiples of 36 and 72 are 72,144, 216,...

**Step 3:** The lowest of the common multiples is the required L.C.M.

72 is the lowest of the common multiples of 36 and 72.

Hence, 72 is the required L.C.M.

**(b) Prime factorization method:**

**e.g.,** find the L.C.M. of 36 and 72.

**Step 1:** Express the given numbers as the product of prime numbers.

36 =2×2×3×3

72 =2×2×2×3×3

**Step 2:** Circle their common prime factors.

36 =2×2×3×3

72 =2×2×2×3×3

Step 2: Circle their common prime factors.

**Step 3:** Consider one set of the common factors and find their product.

2×2×3×3= 36

**Step 4:** Find the product of the remaining factors (encircled factors) and the product obtained in step 3.

36 × 2 = 72

**Step 5:** The product obtained in step 4 is the required L.C.M.

Hence, the L.C.M. of 36 and 72 is 72.

**(c) Division Method:**

**e.g.,** find the L.C.M. of 36 and 72.

**Step 1:** Write the given numbers separated by commas between them.

**Step 2:** Divide the given numbers by a prime factor common to them.

**Note: Start with the least prime factor common to the given numbers.**

**Step 3:** Continue the process until all the factors are prime.

**Step 4:** Find the product of all the prime factors obtained in step 3, which gives the required L.C.M.

2×2×3×3×2= 72

Therefore, 72 is the L.C.M. of 36 and 72.

- C. M. of given numbers = Their H. C. F × The product of the prime factors left encircled.

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