The following real numbers have decimal expansions as given below. In each case, decide

[question] Question. 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 $\frac{\mathbf{P}}{\mathbf{q}}$, what can you say about the prime factors of $\mathrm{g}$ ? (i) $43.123456789$ (ii) $0.120120012000120000 \ldots$ (iii) $43 . \overline{23456789}$ [/question] [solution] Solution: (i) $43.123456789$ Since, the decimal expansion terminates, so the given real number is rational and there...

Prove that the following are irrationals :

[question] Question. Prove that the following are irrationals : (i) $\frac{1}{\sqrt{2}}$ (ii) $7 \sqrt{5}$ (iii) $\mathbf{6}+\sqrt{\mathbf{2}}$ [/question] [solution] Solution: (i) Let us assume, to the contrary, that $\frac{1}{\sqrt{2}}$ is rational. That is we can find coprime integers a and $b(b \neq 0)$ such that, $\frac{1}{\sqrt{2}}=\frac{\mathbf{P}}{q}$ Therefore, $q=\sqrt{\mathbf{z}_{\mathbf{p}}}$ Squaring on both sides, we get $\mathrm{q}^{2}=2 \mathrm{p}^{2}$ ...(1) Therefore, 2 divides...

Find the $\mathrm{LCM}$ and HCF of the following pairs of integers and verify that LCM $\times \mathrm{HCF}=$ product of two numbers.

[question] Question. Find the LCM and HCF of the following pairs of integers and verify that LCM $\times \mathrm{HCF}=$ product of two numbers. (i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54 [/question] [solution] Solution: (i) 26 and 91 So, $26=2 \times 13$ So. $91=7 \times 13$ Therefore, $\operatorname{LCM}(26,91)=2 \times 7 \times 13=182$ $\operatorname{HCF}(26,91)=13$ Verification : LCM $\times$ HCF $=182 \times 13=2366$ and $26 \times 91=2366$ i.e., $\mathrm{LCM} \times \mathrm{HCF}=$ produ...

Express each number as product of its prime factors :

[question] Question. Express each number as product of its prime factors : (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429 [/question] [solution] Solution: (i) 140 So, $140=2 \times 2 \times 5 \times 7=2^{2} \times 5 \times 7$ (ii) 156 So, $156=2 \times 2 \times 3 \times 13=2^{2} \times 3 \times 13$ So, $3825=3^{2} \times 5^{2} \times 17$ (iv) 5005 So, $5005=5 \times 7 \times 11 \times 13$ (v) 7429 So, $7429=17 \times 19 \times 23$ [/solution]...