Real Numbers Test - 3

A real number is an irrational number when it has a non-terminating, non-repeating decimal representation.

85085 = 5 × 7 × 11 × 13 × 19

Thus, there are 5 prime factors in the prime factorization of 85085.

Recall the theorem : Let x be a rational number whose decimal expansion terminates.

Then x can be expressed in the form p /q where p and q are co-prime, and the

prime factorisation of q is of the form 2ⁿ5^m, where n, m are non-negative integers.

1256.1782 is a terminating decimal expansion factors of the denominator can be expressed as 2ⁿ 5^m

It is a product of powers of 2 and 5.

We see that 303 > 161

303 = (161×1) + 142

161 = (142×1) + 19

142 = (19×7) + 9

19 = (9×2) + 1

9 = (1×9) + 0

The divisor in the step where the remainder is 0, is 1.

Hence, the HCF is 1.

A rational number can be expressed as a terminating decimal if the denominator has factors 2 and 5.

smallest composite number = 4

and the smallest even number = 2

Using the Fundamental Theorem of Arithmetic,

2 = 2  and 4 = 2 × 2

So, HCF of 2 and 4 = 2

Given, a = bq + r, 0 ≤ r < b and b = 4

So, r can take values 0, 1, 2, 3.

Thus, r ≠ 4.

119² - 111² = (119 + 111) (119 - 111) = 230 8 (Even number)

Hence, 119² - 111² is a composite number, since a composite number has more than 2 factors other than itself and 1.

If x is non-zero rational number and y is an irrational number,

then the product = x × y

So, the irrational number y will still be a factor of the product, and hence the product will be irrational.