The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , p q what can you say about the prime factors of q?

$left parenthesis iii right parenthesis space space 43. top enclose 123456789$

Solution

∴ 43.123456789123456789... which is non-terminating and repeating. Therefore, it is a rational number which can be expressed in the form p/q, where p and q are co-primes and the prime factors of ‘q’ are of the form 2ⁿ.5ⁿ, where ‘n’ is non-negative integer.

Some More Questions From Real Numbers Chapter

Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

Express each number as a product of its prime factors: (i) 140

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Real Numbers

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , p q what can you say about the prime factors of q?

$left parenthesis iii right parenthesis space space 43. top enclose 123456789$

Some More Questions From Real Numbers Chapter

Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

Express each number as a product of its prime factors: (i) 140

Express each number as a product of its prime factors: (ii) 156

Express each number as a product of its prime factors: (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

Express each number as a product of its prime factors: (iv) 5005

Express each number as a product of its prime factors: (v) 7429

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) 26 and 91

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (ii) 510 and 92

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (iii) 336 and 54

Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21

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