Short Questions
Q1. Write 10 as a sum of consecutive numbers.
Answer:
10 = 1 + 2 + 3 + 4
Step 1: Start from 1
Step 2: Keep adding consecutive numbers
1 + 2 + 3 + 4 = 10
Q2. Express 12 as sum of consecutive numbers.
Answer:
12 = 3 + 4 + 5
Step 1: Try middle numbers
Step 2: 3 + 4 + 5 = 12
Q3. Is 15 expressible as sum of consecutive numbers?
Answer:
Yes
15 = 7 + 8
or 4 + 5 + 6
Step: Try grouping consecutive numbers
Q4. Find parity of 3 + 4 – 5 + 6.
Answer:
3 + 4 – 5 + 6 = 8
8 is even
Q5. Is 4m + 2q always even?
Answer:
Yes
Step 1: 4m is even
Step 2: 2q is even
Step 3: even + even = even
Q6. Find a number leaving remainder 3 when divided by 5.
Answer:
Example: 8
Step: 8 ÷ 5 = remainder 3
Q7. Write general form of numbers leaving remainder 3 when divided by 5.
Answer:
5k + 3
Q8. Check if 123 is divisible by 9.
Answer:
1 + 2 + 3 = 6
6 is not divisible by 9
So, 123 is not divisible by 9
Q9. Check divisibility of 405 by 9.
Answer:
4 + 0 + 5 = 9
9 is divisible by 9
So, 405 is divisible by 9
Q10. Find digital root of 489710.
Answer:
4 + 8 + 9 + 7 + 1 + 0 = 29
2 + 9 = 11
1 + 1 = 2
Digital root = 2
Long Questions
Q1. The sum of four consecutive numbers is 34. Find the numbers.
Answer:
Step 1: Let numbers be
x, x+1, x+2, x+3
Step 2: Form equation
x + (x+1) + (x+2) + (x+3) = 34
Step 3: Simplify
4x + 6 = 34
Step 4: Solve
4x = 28
x = 7
Step 5: Numbers are
7, 8, 9, 10
Q2. Prove that sum of two multiples of 8 is also a multiple of 8.
Answer:
Step 1: Let numbers be
8a and 8b
Step 2: Add them
8a + 8b
Step 3: Factorize
= 8(a + b)
Step 4: Conclusion
Since 8 is a factor → multiple of 8
Q3. Check if 7309 is divisible by 9.
Answer:
Step 1: Add digits
7 + 3 + 0 + 9 = 19
Step 2: Add again
1 + 9 = 10
Step 3: Add again
1 + 0 = 1
Step 4: Conclusion
Not divisible by 9
Q4. Find numbers that leave remainder 2 when divided by 6. Prove sum of three such numbers is divisible by 6.
Answer:
Step 1: General form
6k + 2
Step 2: Take three numbers
(6a+2), (6b+2), (6c+2)
Step 3: Add them
= 6a + 2 + 6b + 2 + 6c + 2
Step 4: Simplify
= 6(a + b + c) + 6
Step 5: Factor
= 6(a + b + c + 1)
Step 6: Conclusion
Always divisible by 6
Q5. Check whether 462 is divisible by 11 using shortcut.
Answer:
Step 1: Alternate signs
4 – 6 + 2
Step 2: Calculate
4 – 6 + 2 = 0
Step 3: Conclusion
0 is multiple of 11
So, 462 is divisible by 11
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