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Playing With Numbers

Prime Factors of a Number

While communicating your credit card information to Amazon, logging into your bank, or sending a manually encrypted email to your colleague, you are constantly using computer encryption. This means that you are are constantly using prime numbers. One of the most common encryption schemes is based on prime numbers. Let us also try to understand the prime factors in detail. Let us do it right now.

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Prime Factors of a Number

Let us first see what factors of given numbers are. What does factorisation mean? The factorisation means writing a number as the product of its factors means in terms of its factors. Take a number say a and find the numbers which divide a. Now, the numbers which divide a  are called factors of a.

Suppose you have a number 40. So this number can be written as 40 = 2 × 20 = 4× 10 = 5 × 8. So we can write 40 as the product of factors in many such ways. From the above example, we can say 2, 4, 5, 20, 10 and 10 are the factors of the number 40 and the number 40 is factorised.

Take a number say a and find the numbers which divide a. Now, the numbers which divide a  are called factors of a. Now again take another number b. Find the factors of b. Now, factors which appear both in a  and b are called common factors. Suppose you take two numbers 6 and 4. Now, 6 = 2 × 3 and 4 = 2 × 2. So, the common factors of 6 and 4 are 2.

Let us have a quick revision of prime numbers. The numbers having factors 1 and number itself are known as prime numbers. Each prime number is only divisible by 1 and itself. That means the number 1 can never be a prime number. So any prime number should have only two factors and it should be greater than 1.

Factors are the numbers you multiply together to get another number

What is Prime Factorisation?

Prime Factors

Let us take a number 60. So, 60 can be written as 1 × 60. But this number 60 can be further split as 1 × 2 × 30. Similarly, we can split 30 as 1 × 2 × 2× 15 = 1 × 2 × 2× 3 × 5.

Now the expression we got above cannot be split further. Every number here is indivisible. The numbers we have obtained are all prime numbers. Also from the above example, we can say that 1 × 2 × 2× 3 × 5 is the prime factorisation of number 60.

Prime factorisation means whenever we express a given number as the product of prime factors than we say that we have done prime factorisation the given number. So suppose you want to write the prime factors of the number 60, then it can be written as 60 = 2 × 2× 3 × 5.

Let us now take another example. Suppose we take 560 as the number.

560 = 2 × 270 = 2 × 2 × 135
= 2 × 2 × 3 × 45
Now,
= 2 × 2 × 3 × 3 ×15
= 2 × 2 × 3 × 3 × 3× 5

So we can write 560  as the product of 2 × 2 × 3 × 3 × 3× 5. And these factors cannot be further divided into smaller factors, we can say that 2, 2, 3, 3, 3, 5 are prime factors of 540.

Suppose we take 70 as the number as 70 = 2 × 35 = 2 × 5 × 7. So we can write 70  as the product of 2 × 5 × 7. And these factors cannot be further divided into smaller factors, we can say that 2, 5, 7 are prime factors of 70.

Solved Examples for You

Question 1: Three common multiples of 18 and 16 are

  1. 18, 6, 9
  2. 18, 36, 6
  3. 36, 54, 72
  4. None

Answer : C is the correct option. The multiple of 18 is 18, 36 and 54. Multiples of 6 are 6, 12 and 18. Here the first common multiple will be 18. the next common multiple will be multiples of 18. The first three common multiples of 18 and 6 are 18, 36 and 54.

Question 2.  1 is a ________ of every prime number.

  1. Factor
  2. Multiple
  3. Both factor and multiple
  4. None of these

Answer : A is the correct option. 1 is a factor of every prime number.

Question 3: How do you find the prime factors of a number?

Answer: The prime factors of a number refer to the entire prime numbers that, when we multiply together will equal the original number. Thus, one can find the prime factorization of a number by making use of a factor tree and further dividing the number into smaller parts.

Question 4: What does factorization mean?

Answer: The term factorization refers to writing a number as the product of its factors means in terms of its factors. Take a number say ‘b’ and find the numbers which divide ‘b.’ At this time, the numbers which divide a  are called factors of ‘b’.

Question 5: What does prime factorization mean?

Answer: Prime factorization refers to whenever you express a given number as the product of prime factors than you say that you have done prime factorization of the given number. Thus, if you wish to write the prime factors of the number 60, then you can write it as 60 = 2 × 2× 3 × 5.

Question 5: What is the prime factorization of 70?

Answer: As you all know that 70 is a composite number. Thus, it will be as follows:

70 = 1 x 70, 2 x 35, 5 x 14, and 7 x 10.

Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70.

Therefore, Prime factorization: 70 = 2 x 5 x 7.

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