Real Numbers: Rational, Irrational Numbers and HCF/LCM

The decimal expansion of $\frac{61}{13}$ is non-terminating because its denominator 13 is not of the form 2^m 5ⁿ, where 0 _^$\leq$ n and m < _^$\infty$.

The decimal expansion of $\frac{91}{9}$ is not terminating because its denominator 9 is not of the form 2^m 5ⁿ, where 0 _^$\leq$ n, m < _^$\infty$.

$\sqrt 2 \times \sqrt 3 = \sqrt 6$, which is an irrational number.

Prime factorisation of 44 = 2 ^_$\times$ 2 ^_$\times$ 11 = 2^{2 _$\times$} 11¹
Prime factorisation of 90 = 2 ^_$\times$ 3 ^_$\times$ 3 ^_$\times$ 5 = 2^{1 _$\times$} 3^{2 _$\times$} 5¹

$\therefore$  LCM (44, 90) = 2^{2 _$\times$} 3^{2 _$\times$} 5^{1 _$\times$} 11¹ = 4 _^$\times$ 9 _^$\times$ 5 _^$\times$ 11 = 1980

The prime factorisation of 1580 = 2 ^_$\times$ 2 ^_$\times$ 5 ^_$\times$ 79

= 2^{2 _$\times$} 5 _^$\times$ 79
So, 2, 5 and 79 appear in the prime factorisation of 1580.

$\frac{126}{45} = \frac{14}{5}$
Since the denominator of $\frac{14}{5}$is 5, which is of the form 2ⁿ 5^m, where n = 0 and m = 1, so $\frac{126}{45}$ has a terminating decimal expansion.

Consider $\sqrt p$ and $\sqrt q$. If p and q are prime numbers, then $\sqrt p \times \sqrt q$ will be irrational.
^{_$\therefore$}  Product of $\sqrt 5$ and $\sqrt 7$ is an irrational number.

The prime factorisation of 410 = 2^{1 _$\times$} 5^{1 _$\times$} 41
The prime factorisation of 96 = 2^{5 _$\times$} 3

HCF = Product of the smallest powers of each common prime factor in the numbers
HCF (410, 96) = 2¹ = 2

In $\frac{1}{10}$, the denominator is 10, which is in the form 2ⁿ 5^m, where n = 1 and m = 1.
Therefore, the number $\frac{1}{10}$has a terminating decimal expansion.
Also, $\frac{1}{10}$= 0.1, which is terminating.

The prime factorisation of 2345 = 5 ^_$\times$ 7 ^_$\times$ 67

So, the prime numbers which appear in the prime factorisation of 2345 are 5, 7 and 67.

The product of a rational and an irrational number is irrational.
And Difference of rational and irrational is irrational.
Therefore,  $3\sqrt 2 -1$ is an irrational number.

L.C.M. ^_$\times$ H.C.F. = Product of two numbers
720 ^_$\times$ H.C.F. = 240 ^_$\times$ 360
H.C.F. = $\frac{240 \times 360}{720}$
          = 120
Alternatively,
H.C.F. = Product of the smallest powers of each common prime factor in the numbers

The prime factorisation of 240 = 2^{4 _$\times$} 3^{1 _$\times$} 5¹
The prime factorisation of 360 = 2^{3 _$\times$} 3^{2 _$\times$} 5¹
$\therefore$ H.C.F. = 2^{3 _$\times$} 3 _^$\times$ 5 = 8 _^$\times$15 = 120

The denominator of rational number $\frac{1}{19}$ is 19, which is not of the form 2ⁿ 5^m (0 _^$\leq$ n, m < _^$\infty$).
So, the rational number $\frac{1}{19}$ has a non-terminating, non-repeating decimal expansion.
$\frac{1}{19}$= 0.05263157894…….

M = 77 ^_$\times$ 144 ^_$\times$ 45
= 7 ^_$\times$ 11 ^_$\times$ 12 ^_$\times$ 12 ^_$\times$ 9 ^_$\times$ 5
= 7 ^_$\times$ 11 ^_$\times$ 2 ^_$\times$ 2 ^_$\times$ 3 ^_$\times$ 2 ^_$\times$ 2 ^_$\times$ 3 ^_$\times$ 3 ^_$\times$ 3 ^_$\times$ 5
So, prime factors of M are 2, 3, 5, 7 and 11.

If 'a' is an irrational number, then $\frac{a}{a} = \frac{1}{1}$, which is a rational number.

If a number has a non-terminating and non-repeating decimal expansion, then the number must be irrational. So,$\sqrt 7$ is an irrational number that has a non-terminating and non-repeating decimal expansion. 

The prime factorisation of 153 = 3 $\times$ 3 $\times$ 1

The prime factorisation of 4913 = 17 $\times$ 17 $\times$ 17
H.C.F (153, 4913) = product of the smallest powers of each common prime factor in the numbers.
            = 17

M = 44 ^_$\times$ 65 $\times$ 84
M = 4 ^_$\times$ 11 ^_$\times$ 13 ^_$\times$ 5 ^_$\times$ 12 ^_$\times$ 7
= 2 ^_$\times$ 2 ^_$\times$ 11 ^_$\times$ 13 ^_$\times$ 5 ^_$\times$ 2 ^_$\times$ 2 ^_$\times$ 3 ^_$\times$ 7
So, the prime factors of M are 2, 3, 5, 7, 11 and 13.

$\frac{1}{\sqrt 2}$ has a non-terminating repeating decimal expansion.

Irrational numbers have non-terminating, non-repeating decimal expansion.
The decimal expansion is 0.132013200132000132..…, which is non-terminating and non-repeating.

The prime factorisation of 49 = 7²
1 _^$\times$ 17¹
The prime factorisation of 147 = 3^{1 _$\times$} 7²
L.C.M = Product of the greatest powers of each prime factor involved in the numbers.
 = 7^{2 _$\times$} 17 _^$\times$ 3 = 2499

The prime numbers occurring in prime factorisation of 58785 are 5 and 3

$\frac{\sqrt 5 \times \sqrt 2}{\sqrt {10}} = \frac{\sqrt{10}}{\sqrt{10}}$= 1, so it is a rational expression.

The prime factorisation of 95 = 5 $\times$ 19
The prime factorisation of 323 = 17 $\times$ 19
The prime factorisation of 551 = 19 $\times$ 2

H.C.F = Product of the smallest powers of each common prime factor in the numbers.
H.C.F (95, 323, 551) = 19

The prime numbers which appear in the prime factorisation of 13720 are 2, 5 and 7.

$\sqrt 6 \times \sqrt 3 = \sqrt {18}$= 3$\sqrt 2$ is an irrational expression.

The irrational numbers have non-terminating and non-repeating decimal expansions.
So $\sqrt 3$, which is an irrational number, has non-terminating and non-repeating decimal expansion.

The prime factorisation of 119 = 7¹ $\times$ 17
The prime factorisation of 343 = 7³3 $\times$ 17 = 5831.

The prime numbers which appear in the prime factorisation of 1768 are 2, 13 and 17

The prime factorisation of 36 = 2^{2 _$\times$} 3²
The prime factorisation of 24 = 2^{3 _$\times$} 3
The prime factorisation of 72 = 2^{3 _$\times$} 3²
2 _^$\times$ 3 = 4 _^$\times$ 3 = 12

If the decimal expansion of a number is non-terminating and non-repeating, then it is an irrational number.
So, 0.1234569873215... is non-terminating and non-repeating, and hence, it is an irrational number. 

Prime numbers occurring in the prime factorisation of 5829 are 3, 29 and 67

$\frac{- \sqrt {2}}{\sqrt {2}} = -1$is not an irrational number.

The denominator of $\frac{1}{6}$ is 6, which is not of the form 2ⁿ 5^m (0 _^$\leq$ n, m < _^$\infty$). So its decimal expansion is non-terminating and repeating.
$\frac{1}{6}= 0.1666...$

The prime factorisation of 375 = 3 ^_$\times$ 5³
The prime factorisation of 2500 = 2^{2 _$\times$} 5⁴
3 _$\times$ 5⁴ _$\times$ 2² = 7500

$\sqrt 2 \times \sqrt 8 = \sqrt {16} = 4$is an integer and all the integers are rational numbers.
So, $\sqrt 2 \times \sqrt 8$ is a rational number as well as an integer.

M = 34 ^_$\times$ 65 ^_$\times$ 133
M = 2 ^_$\times$ 17 ^_$\times$ 13 ^_$\times$ 5 ^_$\times$ 19 ^_$\times$ 7
    = 2 ^_$\times$ 5 ^_$\times$ 7 ^_$\times$ 13 ^_$\times$ 17 ^_$\times$ 19

Let x be the second number.
H.C.F. ^_$\times$ L.C.M. = Product of two numbers
12 ^_$\times$ 240 = 60 ^_$\times$ x
$\Rightarrow$ x = $\frac{240 \times 12}{60}$ = 48

M = 33 ^_$\times$ 221 $\times$ 203
M = 3 ^_$\times$ 11 ^_$\times$ 13 ^_$\times$ 17 ^_$\times$ 29 ^_$\times$ 7
The product of prime factors of M = 3 ^_$\times$ 7 ^_$\times$ 11 ^_$\times$ 13 ^_$\times$ 17 ^_$\times$ 29

Irrational numbers have non-terminating and non-repeating decimal expansions.
$\pi$and e are irrational numbers. So, they have non-terminating and non-repeating decimal expansions.