All questions of Number Play for Class 8 Exam

Power of 2 like 4,6,8,12,32 can cot be expressed as a sum of consecutive no.

Understanding Co-prime Numbers
Co-prime numbers, also known as relatively prime numbers, are pairs of numbers that have no common positive factors other than 1. This means their greatest common divisor (GCD) is 1.
Analysis of Each Pair
Let's evaluate each pair to determine if they are co-prime:
a) 6 and 8
- Factors of 6: 1, 2, 3, 6
- Factors of 8: 1, 2, 4, 8
- Common Factor: 2
- GCD: 2
- Conclusion: Not co-prime
b) 9 and 15
- Factors of 9: 1, 3, 9
- Factors of 15: 1, 3, 5, 15
- Common Factor: 3
- GCD: 3
- Conclusion: Not co-prime
c) 8 and 25
- Factors of 8: 1, 2, 4, 8
- Factors of 25: 1, 5, 25
- Common Factor: 1
- GCD: 1
- Conclusion: Co-prime
d) 12 and 18
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Common Factors: 1, 2, 3, 6
- GCD: 6
- Conclusion: Not co-prime
Final Conclusion
Among the pairs listed, only 8 and 25 are co-prime. Their only common factor is 1, making them relatively prime to each other. Thus, the correct answer is option 'C'.

Understanding the Problem
To determine which expression is always divisible by 2 when n is an integer, we analyze the four options provided.
Options Analysis
- Option A: n² + n
- This expression can be factored: n(n + 1).
- Here, n and n + 1 are two consecutive integers.
- One of these two consecutive integers is always even, making their product (n(n + 1)) always divisible by 2.
- Option B: n² + 1
- This expression is not guaranteed to be even. For example:
- If n = 1: 1² + 1 = 2 (even)
- If n = 2: 2² + 1 = 5 (odd)
- Thus, this expression is not always divisible by 2.
- Option C: n² - 1
- This expression can be factored as (n - 1)(n + 1).
- While (n - 1) and (n + 1) are consecutive integers, making their product even, it will not always be divisible by 2 depending on the parity of n.
- For instance, if n = 2: 2² - 1 = 3 (odd).
- Option D: n³
- This expression is divisible by 2 if n is even but not if n is odd.
- Example: If n = 1: 1³ = 1 (odd); if n = 2: 2³ = 8 (even).
Conclusion
The only expression that is always divisible by 2 for any integer n is:
Option A: n² + n, since it involves the product of two consecutive integers, ensuring at least one of them is always even.

Understanding Parity Rules
In mathematics, particularly in number theory, the concept of parity refers to whether an integer is even or odd.
Definitions:
- Even Number: An integer that is divisible by 2 (e.g., -4, -2, 0, 2, 4).
- Odd Number: An integer that is not divisible by 2 (e.g., -3, -1, 1, 3).
Multiplication of Even and Odd Numbers:
When we multiply an even number by an odd number, the result is always even.
Explanation:
- An even number can be represented as 2n, where n is an integer.
- An odd number can be represented as 2m + 1, where m is also an integer.
Now, when you multiply them:
- Even × Odd = (2n) × (2m + 1)
- Distributing gives: 2n * 2m + 2n * 1 = 4nm + 2n
The result, 4nm + 2n, is clearly divisible by 2, confirming that the product is even.
Conclusion:
Hence, the result of multiplying an even number by an odd number is always an even number. Therefore, the correct answer to the question "What is the result of Even × Odd?" is:
Answer: Even (Option A)

Understanding Divisibility by 11
To determine if a number is divisible by 11, you can use the divisibility rule for 11. This rule states that a number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is either 0 or a multiple of 11.
Step-by-Step Analysis of Each Option
1. Option A: 1001
- Odd position digits: 1 (1st) + 0 (3rd) = 1
- Even position digits: 0 (2nd) + 1 (4th) = 1
- Difference: |1 - 1| = 0
- Since 0 is divisible by 11, 1001 is divisible by 11.
2. Option B: 1234
- Odd position digits: 1 (1st) + 3 (3rd) = 4
- Even position digits: 2 (2nd) + 4 (4th) = 6
- Difference: |4 - 6| = 2
- 2 is not divisible by 11, so 1234 is not divisible by 11.
3. Option C: 2456
- Odd position digits: 2 (1st) + 5 (3rd) = 7
- Even position digits: 4 (2nd) + 6 (4th) = 10
- Difference: |7 - 10| = 3
- 3 is not divisible by 11, so 2456 is not divisible by 11.
4. Option D: 3246
- Odd position digits: 3 (1st) + 4 (3rd) = 7
- Even position digits: 2 (2nd) + 6 (4th) = 8
- Difference: |7 - 8| = 1
- 1 is not divisible by 11, so 3246 is not divisible by 11.
Conclusion
Based on the analysis, the only number among the options that is divisible by 11 is 1001.

Understanding the Problem
To find how many expressions can be formed using + or - signs between the three consecutive numbers n, n+1, and n+2, we need to analyze the options available for the signs.
Available Signs
1. We have three numbers: n, n+1, n+2.
2. We can place either a + or a - sign between each pair of numbers.
Setting Up the Expressions
- There are two gaps between the three numbers:
- Between n and n+1
- Between n+1 and n+2
Calculating Combinations
- For each gap, we have two choices:
- First gap (between n and n+1): either + or -
- Second gap (between n+1 and n+2): either + or -
This means we can calculate the total combinations of signs as follows:
- For the first gap: 2 options (either + or -)
- For the second gap: 2 options (either + or -)
Total Expressions
- Using the multiplication principle, the total number of expressions can be calculated as:
Total expressions = 2 (first gap) * 2 (second gap) = 4
Conclusion
Thus, the total number of distinct expressions that can be formed using + or – signs between the three consecutive numbers n, n+1, and n+2 is 4.
The correct answer is option A.

Understanding the Options
To determine which expression is always odd, let's analyze each option provided:
Option A: 2n + 2
- This expression can be factored as 2(n + 1).
- Since n is an integer, (n + 1) is also an integer, making the entire expression even.
Option B: n² + n
- This can be factored as n(n + 1).
- Regardless of whether n is odd or even, one of these factors will always be even, resulting in an even product.
Option C: 2n + 1
- Here, 2n is even (since it is a multiple of 2).
- Adding 1 to an even number always yields an odd number.
- Therefore, this expression is always odd for any integer value of n.
Option D: 4n
- This can be factored as 4 times n.
- Since 4 is even, this expression is also even for any integer n.
Conclusion
Among the options analyzed, only Option C: 2n + 1 is always odd. This is because it consists of an even number (2n) plus one, which guarantees the result will always be odd, regardless of the integer value of n.
Thus, the correct answer is indeed Option C.

Understanding Divisibility Rules
To determine the divisibility of numbers based on the sum of their digits, we can apply specific rules. In this case, we focus on the sum being 27.
Divisibility by 9
- A number is divisible by 9 if the sum of its digits is divisible by 9.
- Since the sum of the digits is 27, we can check:
- 27 ÷ 9 = 3, which is an integer.
- Thus, any number with a digit sum of 27 is divisible by 9.
Divisibility by 6
- A number is divisible by 6 if it is divisible by both 2 and 3.
- While a sum of 27 indicates divisibility by 3, we cannot conclude anything about evenness (divisibility by 2) without knowing the last digit.
- Therefore, we cannot confirm divisibility by 6.
Divisibility by 11
- A number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum in even positions is either 0 or divisible by 11.
- The sum of the digits alone (27) does not provide enough information to check this rule.
Divisibility by 7
- The divisibility rule for 7 involves more complex calculations or specific checks on the number itself.
- Again, without the actual number, we cannot ascertain divisibility by 7.
Conclusion
In summary, the only definitive conclusion we can draw from a digit sum of 27 is that the number is divisible by 9. Thus, option 'B' is the correct answer.