Prime factors, lcm and hcf

Prime factors of numbers.

Prime factors of a number are the factors of that number which are not divisible by any other number except that number and 1. For example, the prime factors of 30 are 2,3 and 5 because out of all the factors of 30 (1,2,3,5,6,10,15 and 30), it is only 2,3 and 5 that are not divisible by any other number except themselves and 1.

Examples:

Write the factors of the following numbers and state the prime factors, hence express each number as a product of its prime factors.

• 22 (b) 50              (c) 63               (d) 42

Solutions:                       

• 22

Factors of 22 are 1, 2, 11, 22

The prime factors of 22 are 2 and 11

Product of its prime factors of 22 = 2 x11

• Factors of 50 are 1, 2, 5, 10, 25 and 50

The prime factors of 50 are 2 and 5

Product of its prime factors of 50 = 2 x 5 x 5 = 2 x 5²

• Factors of 63 are 1, 3, 7, 9, 21 and 63

The prime factors of 63 are 3 and 7

Product of its prime factors of 63 = 3 x 3 x 7 = 3² x 7

• Factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42

The prime factors of 42 are 2, 3 and 7

Product of its prime factors of 42 = 2 x 3 x 7

THE LEAST COMMON MULTIPLES (LCM) OF NUMBERS

The LCM of two or more quantities is the smallest common multiple they have.

For example:

The multiple of 2: 2, 4, 6, 8, 10, 12…

The multiple of 3: 3, 6, 9, 12, 15…

Notice that 6 and 12 are the common multiples of 2 & 3

But the Least Common multiple (LCM) is 6.

Example: Find the LCM of 22, 30 and 40

Method: Express each number as a product of its prime factors.

22 = 2 x 11

30 = 2 x 3 x 5

40 = 2 x 2 x 2 x5 = 2³ x 5

The prime factors of 22, 30 and 40 are 2, 3, 5, and 11.

The highest power of each prime factor must be in the LCM.

These are 2³, 3, 5 and 11

Thus, LCM = 2³ X 3 X 5 X 11

= 8 X 3 X 5 X 11

= 1320

PRACTICE QUESTIONS

• Find the L.C.M of the following: a) 12, 15 and 18      b) 20, 28 and 30  c) 216 and 168
• Find the HCF of the following: a) 72, 108 and 54 b) 36, 54 and 60  c) 54 and 105
• Find the factors of the square roots of the following:

1. 1 936 b) 2 025           c) 2 916           d) 3 136           e) 1 225

• What must the multiplied to 240 to make it a perfect square?
• Express 11 664 as a product of its factors in index form

ASSIGNMENTS

• Using the product of prime factors, find the square root of i) 400 ii) 750
• What is the smallest number by which 90 must be multiplied to obtain a perfect square?
• Find the square root of the following by factorization method

1. 5 184 ii) 46 565         iii) 7 744         d) 2 916           e) 41 209

• A square plank has an area of 116.64cm². What is the length of its sides?
• A pile of nuts can be shared equally between 8 boys, between 9 boys and between 10 boys. What is the least number of nuts in the pile?
• How many five-digit numbers are divisible by 11, If the first digit is 1, the third digit is 3 and the fifth digit is 5?
• Express 38 as a product of two prime numbers.
• Factorize and so find the square root of 64, 81, 144, 256, 729, 1600