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Every counting number has an infinite number of (a) Factors (b) Multiples (c) Prime factors (d) None of these

Every counting number has an infinite number of
(a) Factors
(b) Multiples
(c) Prime factors
(d) None of these

Grade:12th pass

1 Answers

Pawan Prajapati
askIITians Faculty 9723 Points
20 days ago
Here, we need to choose an option to complete the statement. We will use the definitions and examples of the given terms to find the correct option that can complete the given statement. Complete step by step solution: We will use the definitions and examples of the given terms to find the correct option that can complete the given statement. If a number is divided by another number such that the remainder is 0, then the second number is called a factor of the first number. For example: When 30 is divided by 6, the quotient is 5 and the remainder is 0. Thus, we can say that 6 is a factor of 30. The quotient 5 is also a factor of 30. Let us take the natural number 24. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Therefore, we can see that the number 24 does not have an infinite number of factors. Thus, option (a) is incorrect. Now, we will define what a multiple is. When a number is multiplied by a natural number, then the product is called the multiple of the first number. For example: The first five natural numbers are 1, 2, 3, 4, 5. The product of 9 and 1 is 9, the product of 9 and 2 is 18, the product of 9 and 3 is 27, the product of 9 and 4 is 36, and the product of 9 and 5 is 45. Therefore, we get the first five multiples of 9 as 9, 18, 27, 36, and 45. Similarly, for every natural number, the multiple of 9 is different. Since there are infinite natural numbers, there are infinite multiples of 9. Therefore, every counting number has an infinite number of multiples. Thus, the correct option is option (b). Note: We used the term ‘natural numbers’ in the solution. Natural numbers include all the positive integers like 1, 2, 3, etc. For example: 1, 2, 100, 400, 52225, are all natural numbers. The natural numbers are the basic numbers we use when counting objects. We will check the third option also. A prime factor is a factor of a number which is divisible by 1 and itself only. Let us take the natural number 24. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Here, the factors that are prime numbers are 2 and 3. Therefore, we can see that the number 24 does not have an infinite number of prime factors. Thus, option (c) is incorrect.

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