1. HCF of 8, 9, 25 is
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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
2. The product of a rational and irrational number is
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.
3. The product of three consecutive positive integers is divisible by
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
4. The set A = {0,1, 2, 3, 4, …} represents the set of
The set includes:
• 0
• All positive integers (1, 2, 3, 4, …)
This is the definition of the set of whole numbers.
5. Which number is divisible by 11?
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
6. LCM of the given number ‘x’ and ‘y’ where y is a multiple of ‘x’ is given by
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.
7. The largest number that will divide 398,436 and 542 leaving remainders 7,11 and 15 respectively is
Find numbers after subtracting the remainders (these are divisible by the required divisor):
1. 398−7=391
2. 436−11=425
3. 542−15=527
Now find the HCF of 391, 425, 527
• 391=17×23
• 425=17×25
• 527=17×31
Common factor = 17.
Ans: A) 17.
8. There are 312, 260 and 156 students in class X, XI and XII respectively. Buses are to be hired to take these students to a picnic. Find the maximum number of students who can sit in a bus if each bus takes equal number of students
Let's find the GCD of 312, 260, and 156.
Find the prime factorization of each number.
• 312 = 2^3 × 3 × 13
• 260 = 2^2 × 5 × 13
• 156 = 2 ^ 2 × 3 × 13
Identify the common factors.
• The common prime factors are 2^2 x 13.
Multiply the common prime factors.
• 2^2×13 = 4×13 = 52
Thus, the maximum number of students who can sit in a bus is 52.
The correct answer is: (a) 52.
9. If b = 3, then any integer can be expressed as a =
If b=3, then any integer a can be expressed in the form: a=3q, 3q+1, or 3q+2
This is because when any integer is divided by 3, the remainder can be 0, 1, or 2. These are the three possible remainders when dividing by 3.
• a=3q (if the remainder is 0),
• a=3q+1 (if the remainder is 1),
• a=3q+2 (if the remainder is 2).
Thus, the correct answer is:
(a) 3q, 3q + 1, 3q + 2.
10. The sum of two irrational numbers is always
The sum of two irrational numbers, in some cases, will be irrational. However, if the irrational parts of the numbers have a zero sum (cancel each other out), the sum will be rational.
For example:
• sqrt 2+(2− sqrt 2)=2 , where the irrational parts (sqrt2) cancel each other, leaving a rational result.
In other cases, where the irrational parts don't cancel, the sum remains irrational, like:
• sqrt{2} + sqrt{3}, which is irrational.
Thus, the sum can be rational or irrational, depending on whether the irrational components cancel out.
11. The product of two different irrational numbers is always
The sum of a rational number and an irrational number is always irrational. This is because the irrational part cannot be "cancelled out" by a rational number
12. The sum of a rational and irrational number is
The sum of a rational number and an irrational number is always irrational. This is because the irrational part cannot be "cancelled out" by a rational number.