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Factors and Multiples are some of the most important concepts of Numbers. Factors and Multiples are related to each other and are so generally studied together. A factor of a number is an exact divisor of that number while a multiple of a number is a number obtained by multiplying it by a natural number.
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Key Terms: Factors, Multiples, Prime Numbers, Number of Factors, Number of Multiples, Prime factorization
Also Check: Difference Between Fraction and Rational Numbers
Factors and Multiples Meaning
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FACTORS: A factor of a number is an exact divisor of that number. In other words, a factor of a number is that number that completely divides the number without leaving a remainder.
Each of the numbers 1, 2,3,4,6, and 12 is a factor of 12. However, none of the numbers 5, 7, 8, 9, 19, and 11 is a factor of 12.
The number of factors of a number is defined. That is the number of a factor of the number 12 is 6.
MULTIPLES: A multiple of a number is a number obtained by multiplying it by a natural number.
If we multiply 3 by 1, 2, 3, 4, 5, and 6 we get,
3, 6, 9,12,15,18 respectively are multiples of 3.
The number of multiples of a number is not defined. There can be infinite multiples of a number.
Examples
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A factor divides the mother number exactly and leaves no remainder.
The factors of 24 are 1, 2, 3, 4, 8, 6, 12, and 24.
We find that if we divide 24 by 1, 2, 3, 4, 8, 6, 12, and 24 the remainder will be 0. So the numbers exactly divide the number 24 and are thus the factors.
A multiple of a number is a number obtained by multiplying it by a natural number.
If we multiply 7 by 1, 2, 3, 4, 5, and 6 we get,
7, 14, 21, 28, 35, 28 respectively are multiples of 7.
Also Check: Natural Numbers and Whole Numbers
Properties
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- 1 is a factor of every number.
We have,
1 = 1*1
10 = 1*10
341 = 1*341
Thus, 1 is a factor of every number.
- Every number is a factor of itself.
We have,
16 = 16*1
72= 72*1
1 = 1*1
- Every factor of a number is an exact divisor of that number.
The factors of 24 are 1, 2, 3, 4, 8, 6, 12, and 24.
We find that if we divide 24 by 1, 2, 3, 4, 8, 6, 12, and 24 the remainder will be 0.
- Every factor of a number is less than or equal to the number.
- of 34 are 1, 2, 17, 34, and 34 is the largest factor and all other factors are less than 34.
- Factors of a given number are finite.
Since every factor of a number is less than or equal to the number. Thus, factors of a given number are finite.
- Every multiple of a number is greater than or equal to the number.
If we multiply 7 by 1, 2, 3, 4, 5, and 6 we get,
7, 14, 21, 28, 35, 28 respectively are multiples of 7. The smallest number is 7.
Thus other multiples are greater than 7.
- Every number is multiple of itself.
- The number of multiples of a given number is infinite.
If we multiply 3 by 1, 2, 3, 4, 5, and 6 we get,
3, 6, 9,12,15,18 respectively are multiples of 3. It is a never-ending list and hence the multiples of a number are infinite.
Also Check: Fractions
Things to Remember
- If one number divides the second number completely, then the first number is said to be a factor of the second number.
- Factors are also called sub-multiples or divisors.
- Each of the numbers 1, 2,3,4,6, and 12 is a factor of 12. However, none of the numbers 5, 7, 8, 9, 19, and 11 is a factor of 12.
- A multiple of a number is a number obtained by multiplying it by a natural number.
- If we multiply 3 by 1, 2, 3, 4, 5, and 6 we get 3, 6, 9,12,15,18 respectively are multiples of 3.
Also Check: Properties of Real Numbers
Sample Questions
Question: Write all the factors of 64? (2 marks)
Answer: The given number is 64 and it is divisible by 1, 2, 4, 8. so the factors of 64 are: 1, 2, 4, 8, 16, 32, and 64.
Question: Write the first six multiples of 25? (2 marks)
Answer: To obtain the first six multiples of 25, we multiply it by 1, 2, 3, 4, 5, and 6 respectively.
We have 25 × 1 = 25
25 × 2 = 50
25 × 3 = 75
25 × 4 = 100
25 × 5 = 125
25 × 6 = 150
The first six multiples of 25 are 25, 50, 75, 100, 125, and 150.
Question: Question: Write all the factors of 20? (2 marks)
Answer: The given number is 20 and it is divisible by 1, 2, 4, 5. so the factors of 20 are: 1, 2, 4, 5, 10, and 20.
Question: How many factors are there of 20? (2 marks)
Answer: The factors of 20 are: 1, 2, 4, 5, 10, and 20. So, the number of factors of 20 is 6.
Question: Is there any natural number having no factor at all? (2 marks)
Answer: No, each natural number has at least one factor. Even the number 1 is a factor in itself.
Question: Write the first three multiples of 15? (2 marks)
Answer: To obtain the first three multiples of 15, we multiply it by 1, 2, and 3 respectively.
We have 15 × 1 = 15
15 × 2 = 30
15 × 3 = 45
The first three multiples of 15 are 15, 30, and 45.
Question: Without actually dividing show that 13 is a factor of 130013? (2 marks)
Answer: We have,
130013 = 130000 + 13
= 13 × (10000 + 1)
= 13 × 10001
13 is a factor of 13 × 10001. Hence, 13 is a factor of 130013.
Question: Find two numbers whose difference is 2 and the product is 24? (2 marks)
Answer: We have to find two factors of 24 whose difference is 2 and the product is 24.
Factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
4 and 6 are two factors of 24 such that their difference is 2 and the product is 24.
Question: The product of the two numbers is 48. Their sum is 19. What are the numbers? (2 marks)
Answer: The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
3 and 16 are the factors whose sum is 19.
Hence, the required numbers are 3 and 16.
Question: Find all the factors and first four multiples of 26 (2 marks)
Answer: The given number is 26 and it is divisible by 1, 2, and 13. So the factors of 64 are 1, 2, 13, and 26.
To obtain the first four multiples of 26, we multiply it by 1, 2, 3, and 4 respectively.
We have 26 × 1 = 26
26 × 2 = 52
26 × 3 = 78
26 × 4 = 104
The first four multiples of 26 are 26, 52, 78, and 104.
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