Question 1

Given that m is the HCF of 98 and 28, and n is the LCM of 98 and 28, determine the value of the expression n - 7m.

Accepted Answer

Answer: 98. The prime factorization of 98 is 2 * 7^2 and of 28 is 2^2 * 7. Therefore, HCF(m) = 14 and LCM(n) = 196. The expression n - 7m evaluates to 196 - 7(14) = 196 - 98 = 98.

Question 2

Let x be the LCM of 4, 6, and 8. Let y be the LCM of 3, 5, and 7. If p is the LCM of x and y, which of the following relationships is correct?

Accepted Answer

Answer: p = 35x. x = LCM(4,6,8) = 24. y = LCM(3,5,7) = 105. p = LCM(x,y) = LCM(24,105) = 840. Checking the options, 35x = 35 * 24 = 840, which equals p.

Question 3

Consider the following statements:
Assertion (A): For any two prime numbers p and q, their HCF is 1 and their LCM is p + q.
Reason (R): For any two natural numbers, the product of their HCF and LCM is equal to the product of the numbers.
Which of the following is correct?

Accepted Answer

Answer: Assertion (A) is false but Reason (R) is true.. Assertion (A) is false because the LCM of two distinct prime numbers p and q is their product (p*q), not their sum (p+q). Reason (R) is true as it states the fundamental relationship between HCF, LCM, and the numbers themselves.

Question 4

Find the largest number that divides 70 to leave a remainder of 5 and divides 125 to leave a remainder of 8.

Accepted Answer

Answer: 13. The required number must exactly divide (70-5)=65 and (125-8)=117. The HCF of 65 (5x13) and 117 (3^2x13) is 13.

Question 5

For any natural number n, which of the given digits cannot be the unit digit of the number 8^n?

Accepted Answer

Answer: 2. The prime factorization of 8^n is (2^3)^n, which only contains the prime factor 2. For a number to have a unit digit of 0, its prime factorization must include both 2 and 5. Therefore, 8^n can never end in 0.

Question 6

If the LCM of two prime numbers p and q, where p > q, is 221, what is the value of the expression 3p - q?

Accepted Answer

Answer: 38. The LCM of two prime numbers is their product. So, p × q = 221. The prime factors of 221 are 17 and 13. Since p > q, we have p = 17 and q = 13. The value of 3p - q is 3(17) - 13 = 51 - 13 = 38.

Question 7

The HCF of 2520 and 6600 is 40, and their LCM is given by 252 × k. Find the value of k.

Accepted Answer

Answer: 1650. Using the property that the product of two numbers is equal to the product of their HCF and LCM: 2520 × 6600 = 40 × (252 × k). Solving for k, we get k = (2520 × 6600) / (40 × 252) = (10 × 6600) / 40 = 6600 / 4 = 1650.

Question 8

What is the Least Common Multiple (LCM) of 850 and 500?

Accepted Answer

Answer: 17 × 500. The prime factorization of 850 is 2 × 5² × 17 and for 500 is 2² × 5³. The LCM is the product of the highest powers of all prime factors, which is 2² × 5³ × 17. This value is equal to 17 × (4 × 125) = 17 × 500.

Question 9

If two positive integers p and q are expressed as p = 18a²b¹ and q = 20a³b², where a and b are prime numbers, what is the LCM of p and q?

Accepted Answer

Answer: 180 a³b². The LCM is found by taking the highest power of each prime factor present in the numbers. p = 2 × 3² × a² × b and q = 2² × 5 × a³ × b². The LCM is 2² × 3² × 5 × a³ × b² = 180a³b².

Question 10

What is the ratio of the HCF to the LCM of the least composite number and the least prime number?

Accepted Answer

Answer: 1:2. The least composite number is 4 and the least prime number is 2. The HCF of 4 and 2 is 2, and their LCM is 4. The ratio of HCF to LCM is 2:4, which simplifies to 1:2.

Question 11

The HCF of two positive numbers is 12 and their product is 6336. How many pairs of such numbers are possible?

Accepted Answer

Answer: 2. Let the numbers be 12a and 12b where a and b are co-prime. Their product is 144ab = 6336, so ab = 44. The co-prime pairs for (a, b) are (1, 44) and (4, 11), resulting in two possible pairs of numbers.

Question 12

For any natural number 'n', with which of the following digits can the number (12)^n not end?

Accepted Answer

Answer: 0. For a number to end with the digit 0, its prime factorization must include both 2 and 5. The prime factorization of 12 is 2^2 * 3, which does not contain the prime factor 5.

Question 13

How can the number 385 be expressed as a product of its prime factors?

Accepted Answer

Answer: 5 x 7 x 11. The prime factorization of 385 is found by dividing it by successive prime numbers. 385 ÷ 5 = 77, and 77 ÷ 7 = 11. Thus, 385 = 5 x 7 x 11.

Question 14

What are the HCF and LCM of the numbers 12, 21, and 15, respectively?

Accepted Answer

Answer: 3, 420. The prime factorization of the numbers are 12=2²×3, 21=3×7, and 15=3×5. The HCF is the product of the lowest powers of common factors, which is 3. The LCM is the product of the highest powers of all prime factors, which is 2²×3×5×7 = 420.

Class 10 Maths: The Fundamental Theorem of Arithmetic — Practice Questions with Answers

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