Explanation: Note : Check > EBook > Math Shortcut method > Find HCF & LCM...... The HCF is the greatest number that divides all the given numbers. Let's find the HCF of 8, 9, and 25: • The factors of 8 are: 1,2,4,8 • The factors of 9 are: 1,3,9 • The factors of 25 are: 1,5,25 The only common factor among 8, 9, and 25 is 1. Thus, the HCF of 8, 9, and 25 is: 1

Explanation: The product of a rational number and an irrational number is always irrational. This is because a rational number cannot "cancel out" the irrationality of another number, and thus the product will remain irrational.

Explanation: The product of three consecutive positive integers is always divisible by 6. Why: • One of the integers will always be divisible by 2 (since every second integer is even). • One of the integers will also be divisible by 3 ( since every third integer is divisible by 3). Therefore, the product of three consecutive integers is always divisible by both 2 and 3, and thus by 6. Ans: 6

Explanation: The set includes: • 0 • All positive integers (1, 2, 3, 4, …) This is the definition of the set of whole numbers.

Explanation: Rule for Divisibility by 11: Take the alternating sum of the digits (i.e., subtract and add digits in alternating positions). If the result is 0 or divisible by 11, then the number is divisible by 11. Let's check : (a) 1516 • Digits: 1, 5, 1, 6 • Alternating sum: 1 − 5 + 1 − 6 = −9 • Not divisible by 11 - Wrong (b) 1452 •, Digits: 1, 4, 5, 2 • Alternating sum: 1 − 4 + 5 − 2 = 0 • 0 is divisible by 11 - Correct (c) 1011 • Digits: 1, 0, 1, 1 • Alternating sum: 1 − 0 + 1 − 1 = 1 • Not divisible by 11 - Wrong (d) 1121 • Digits: 1, 1, 2, 1 • Alternating sum: 1 − 1 + 2 − 1 = 1 • Not divisible by 11 - Wrong

Explanation: If y is a multiple of x, then the Least Common Multiple (LCM) of x and y is simply y. This is because y is already divisible by x, and hence the LCM is the larger number, which is y. So, the correct answer is: (b) y.