Chapter: Number Play (Letter Number Play)
1. Which of the following expressions is always even for any integer values of the variables?
(a) \(x^2 + 2\) (b) \(4m + 2q\) (c) \(6m - 3n\) (d) \(b^2 + 1\)
2. The sum of any two multiples of 8 is always a multiple of
(a) 4 only (b) 8 only (c) both 4 and 8 (d) neither 4 nor 8
3. If a number leaves a remainder 3 when divided by 5, then it can be written in the form
(a) \(5k + 3\) (b) \(3k + 5\) (c) \(5k - 2\) (d) both (a) and (c)
4. Assertion (A): All eight expressions formed by placing + and – signs between any four numbers have the same parity.
Reason (R): Changing the sign of any term changes the value by an even number.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
5. A number is divisible by 9 if and only if
(a) its units digit is 9 (b) the sum of its digits is divisible by 9 (c) it is divisible by 3 (d) its tens digit is a multiple of 9
6. Assertion (A): The digital root of any multiple of 9 is always 9.
Reason (R): The digital root is the remainder when the number is divided by 9.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
7. Which of the following is never true?
(a) Sum of an odd number and an even number is a multiple of 6.
(b) If 8 divides two numbers, then 8 divides their sum.
(c) If a number is divisible by 12, it is divisible by all factors of 12.
(d) All multiples of 7 are divisible by 7.
8. The remainder when 427 is divided by 9 is
(a) 4 (b) 13 (c) 1 (d) 7
9. In the cryptarithm \(PQ \times 8 = RS\), a possible value of PQ is
(a) 12 (b) 13 (c) 15 (d) 16
10. Two even numbers that are not multiples of 4, when added, always give a multiple of
(a) 2 only (b) 4 (c) 8 (d) none of these
11. Write any algebraic expression that is always even. Justify.
12. If a number leaves a remainder 2 when divided by 3 and also when divided by 4, write an algebraic expression for all such numbers.
13. State whether the statement is always true, sometimes true or never true: “If a number is divisible by both 6 and 4, then it is divisible by 24.” Give one example.
14. Using the divisibility rule for 9, check whether 358095 is divisible by 9.
15. What is the digital root of 489710? How is it related to the remainder when divided by 9?
16. Find the smallest multiple of 9 that contains only even digits.
17. Four consecutive numbers sum to 34. Find the numbers.
18. Without actual division, find which of the following are divisible by 9: 123, 405, 8888, 93547.
19. Using the shortcut for 11, check whether 320185 is divisible by 11 and state the remainder if any.
20. Show that if a number is of the form \(5k + 3\), it always leaves remainder 3 when divided by 5. Give two examples.
21. Explain why the sum of the digits of a number and the number itself leave the same remainder when divided by 9.
22. Prove that when any four numbers are combined using + and – signs in all eight possible ways, all resulting expressions have the same parity. Use algebraic reasoning.
23. (Multi-step) A number leaves remainder 2 when divided by 3, remainder 3 when divided by 4 and remainder 4 when divided by 5. Find the smallest such positive number and explain why it is the smallest.
24. Examine the following statements and classify each as Always True, Sometimes True or Never True. Justify each with algebra and one example/non-example:
(i) The sum of two even numbers is always a multiple of 3.
(ii) If a number is divisible by 9, then reversing its digits gives another multiple of 9.
25. Case: Anshu is exploring sums of consecutive numbers. He observes that every odd number can be written as the sum of two consecutive numbers and some numbers can be written in more than one way.
(a) Express 15 as the sum of consecutive numbers in three different ways.
(b) Can every even number be written as the sum of consecutive numbers? Give one example.
(c) Write an algebraic expression that represents all numbers leaving remainder 3 when divided by 5.
(d) Why can 0 not be expressed as the sum of two positive consecutive natural numbers?
26. Case: Students are checking divisibility shortcuts. They notice that place values follow a pattern of 1 more or 1 less than a multiple of 11.
(a) Using the 11-divisibility shortcut, check whether 462 is divisible by 11.
(b) Find the remainder when 7309 is divided by 9 by adding digits repeatedly.
(c) State the condition for a number to be divisible by both 3 and 8 so that it is divisible by 24.
(d) Why does checking divisibility by 4 and 6 not guarantee divisibility by 24? Give a counter-example.
1. (b) \(4m + 2q = 2(2m + q)\) — always even (1 mark)
2. (c) (1 mark)
3. (d) (1 mark)
4. (a) (1 mark)
5. (b) (1 mark)
6. (b) (1 mark)
7. (a) (1 mark)
8. (a) 4 + 2 + 7 = 13 → 1 + 3 = 4 (1 mark)
9. (a) 12 × 8 = 96 (1 mark)
10. (b) (1 mark)
11. \(4m + 2q = 2(2m + q)\) — factor of 2, hence even (2 marks)
12. \(12n + 2\) (LCM of 3 and 4 is 12) (2 marks)
13. Sometimes true. Example: 12 is divisible by 4 and 6 but not by 24. (2 marks)
14. 3+5+8+0+9+5 = 30, 3+0=3 (not divisible by 9) (2 marks)
15. 2 (digital root = remainder when divided by 9, except when remainder is 0 it is 9) (2 marks)
16. 288 (2 marks)
17. Let numbers be \(n, n+1, n+2, n+3\). Sum = \(4n + 6 = 34\) → \(n = 7\). Numbers: 7,8,9,10 (3 marks)
18. Only 405 (sum of digits = 9) (3 marks)
19. Excess = 3 + 1 + 5 = 9; Short = 2 + 8 + 0 = 10; Difference = –1 → remainder 10 or 10 – 11 = –1 (3 marks)
20. Any number \(5k + 3\) divided by 5 gives quotient \(k\) and remainder 3 (3 marks)
21. Because each place value ≡ 1 (mod 9), so number ≡ sum of digits (mod 9) (3 marks)
22. Changing sign of any term changes value by twice that term (even). Hence parity remains same. All 8 expressions have identical parity. (5 marks)
23. Smallest number = LCM(3,4,5) – 1 = 60 – 1 = 59. It is smallest because any smaller positive number satisfying the conditions does not exist. (5 marks)
24. (i) Sometimes true (example 2+4=6, 2+6=8). (ii) Always true (sum of digits unchanged). (5 marks)
25. (a) 15=7+8=4+5+6=1+2+3+4+5 (1)
(b) Yes, 10=1+2+3+4 (1)
(c) \(5k+3\) (1)
(d) Two positive consecutive numbers sum to at least 1+2=3 (1)
26. (a) Not divisible, remainder 10 or –1 (1)
(b) 7+3+0+9=19→1+9=10→1 (1)
(c) Must be divisible by both 3 and 8 (1)
(d) 12 is divisible by 4 and 6 but not by 24 (1)
All questions are answerable from the NCERT chapter text. Reviewed by GFIS faculty.