Real Numbers Set-2

Q1. If $\gcd(p,q)=6$ and $\operatorname{lcm}(p,q)=180$, what is the value of $p\cdot q$?

Q2. Express the repeating decimal $0.7\overline{36}$ as a fraction in lowest terms.

Q3. Using the Euclidean algorithm, find $\gcd(1001,231)$.

Q4. Assertion (A): For positive integers $a,b$, if $\sqrt{a}+\sqrt{b}$ is rational then both $\sqrt{a}$ and $\sqrt{b}$ are rational.
Reason (R): Both $\sqrt{a}$ and $\sqrt{b}$ are algebraic numbers.

Q5. For which positive integers $n$ is the number $\sqrt{\,n+\sqrt{n}\,}$ a rational number?

Q6. Find the HCF of 495 and 385 by applying the Euclidean algorithm.

Q7. Two positive integers have HCF $6$ and LCM $180$. If their sum is $78$, find the two integers.

Q8. Let $a=2^4\cdot3^2\cdot5$ and $b=2^3\cdot3^3\cdot7^2$. The value of HCF$(a,b)$ is

Q9. Given HCF$(a,b)=15$ and LCM$(a,b)=540$ for positive integers $a,b$, the number of ordered pairs $(a,b)$ possible is

Q10. Assertion (A): For a positive integer $n$ which is not a perfect square, $\sqrt{n}$ is irrational.
Reason (R): If $\sqrt{n}$ is irrational, then for any integer $m\neq 0$, $\sqrt{n}+\sqrt{m}$ is irrational.

Q11. Let the HCF of $36$ and $x$ be $6$ and their LCM be $72$. Find the value of $x$.

Q12. Let $m,n$ be positive integers with $\gcd(m,n)=5$ and $\mathrm{lcm}(m,n)=500$. How many ordered pairs $(m,n)$ of positive integers satisfy these conditions?

Q13. A rational number $\dfrac{m}{175}$ is expressed in lowest terms. What is the length of the repetend (period of the repeating block) in its decimal expansion?

Q14. Assertion (A): If a rational number $r$ (in lowest terms) has denominator $n$ not divisible by $2$ or $5$, then its decimal expansion is purely recurring.
Reason (R): Because $\gcd(10,n)=1$, there exists the least positive integer $k$ such that $10^k\equiv 1\pmod{n}$; this $k$ is the length of the repeating period.

Q15. Let $p$ and $q$ be distinct positive integers with $\gcd(p,q)=1$. If $\sqrt{p}+\sqrt{q}$ is a rational number, which of the following must be true?