-
Question 1: Use Euclid's division algorithm to find the HCF of 4052 and 12576.
Solution: $12576 = 4052 \times 3 + 420$; $4052 = 420 \times 9 + 272$; $420 = 272 \times 1 + 148$; $272 = 148 \times 1 + 124$; $148 = 124 \times 1 + 24$; $124 = 24 \times 5 + 4$; $24 = 4 \times 6 + 0$. HCF is 4.
-
Question 2: Prove that $5 - \sqrt{3}$ is an irrational number.
Solution: Assume $5 - \sqrt{3} = a/b$ (rational). Then $\sqrt{3} = 5 - a/b = (5b-a)/b$. RHS is rational, but $\sqrt{3}$ is irrational. Contradiction. Hence $5 - \sqrt{3}$ is irrational.
-
Question 3: Find the largest number which divides 70 and 125, leaving remainders 5 and 8 respectively.
Solution: Numbers are $70-5=65$ and $125-8=117$. HCF(65, 117) = 13. The largest number is 13.
-
Question 4: Find the HCF and LCM of 96 and 404 by the prime factorization method.
Solution: $96 = 2^5 \times 3$, $404 = 2^2 \times 101$. HCF $= 2^2 = 4$. LCM $= 2^5 \times 3 \times 101 = 9696$.
-
Question 5: There is a circular path around a sports field. Sonia takes 18 minutes to drive one round, while Ravi takes 12 minutes. Suppose they start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?
Solution: We need LCM(18, 12). $18 = 2 \times 3^2$, $12 = 2^2 \times 3$. LCM $= 2^2 \times 3^2 = 4 \times 9 = 36$ minutes.
-
Question 6: Prove that $\sqrt{5}$ is irrational.
Solution: Let $\sqrt{5} = a/b$. $5 = a^2/b^2 \Rightarrow a^2 = 5b^2$. So 5 divides $a$. Let $a=5c$. $(5c)^2 = 5b^2 \Rightarrow 25c^2 = 5b^2 \Rightarrow b^2 = 5c^2$. So 5 divides $b$. Common factor 5 contradicts co-prime assumption.
-
Question 7: Prove that $3 + 2\sqrt{5}$ is irrational.
Solution: Let $3 + 2\sqrt{5} = a/b$ (rational). Then $\sqrt{5} = \frac{1}{2}(\frac{a}{b} - 3)$. Since $a, b$ are integers, RHS is rational. But $\sqrt{5}$ is irrational. Contradiction.
-
Question 8: Show that any positive odd integer is of the form $4q+1$ or $4q+3$, where $q$ is some integer.
Solution: Let $a$ be a positive odd integer. By division algorithm with $b=4$, $a = 4q+r$, $0 \le r < 4$. Possible remainders: 0, 1, 2, 3. $4q$ and $4q+2$ are even. So odd integers are $4q+1$ or $4q+3$.
-
Question 9: Three bells toll at intervals of 9, 12, 15 minutes respectively. If they start tolling together, after what time will they next toll together?
Solution: LCM(9, 12, 15). $9=3^2, 12=2^2 \times 3, 15=3 \times 5$. LCM $= 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180$ minutes.
-
Question 10: If the HCF of 65 and 117 is expressible in the form $65m - 117$, find the value of $m$.
Solution: HCF(65, 117) = 13. $13 = 65m - 117 \Rightarrow 65m = 130 \Rightarrow m = 2$.