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Solutions

Overview

Step-by-step NCERT solutions for Number Play (Chapter 5, NCERT Class 8 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Number Play textbook chapter.

Solved

What these solutions cover

All 32 questions in Number Play are solved in the PDF. Here's what's inside, exercise by exercise:

What Remains?

  1. The sum of four consecutive numbers is 34. What are these numbers?
  2. Suppose p is the greatest of five consecutive numbers. Describe the other four numbers in terms of p.
  3. For each statement below, determine whether it is always true, sometimes true, or never true. Explain your answer. Mention examples and non-examples as appropriate. Justify your claim using algebra.
    • (i) The sum of two even numbers is a multiple of 3.
    • (ii) If a number is not divisible by 18, then it is also not divisible by 9.
    • (iii) If two numbers are not divisible by 6, then their sum is not…
  4. Find a few numbers that leave a remainder of 2 when divided by 3 and a remainder of 2 when divided by 4. Write an algebraic expression to describe all such numbers.
  5. "I hold some pebbles, not too many, When I group them in 3's, one stays with me. Try pairing them up — it simply won't do, A stubborn odd pebble remains in my view. Group them by 5, yet one's still around, But grouping by seven, perfection is found. More than one hundred would be far too bold, Can you tell me the number of pebbles I hold?"
  6. Tathagat has written several numbers that leave a remainder of 2 when divided by 6. He claims, "If you add any three such numbers, the sum will always be a multiple of 6." Is Tathagat's claim true?
  7. When divided by 7, the number 661 leaves a remainder of 3, and 4779 leaves a remainder of 5. Without calculating, can you say what remainders the following expressions will leave when divided by 7? Show the solution both algebraically and visually.
    • (i) 4779 + 661
    • (ii) 4779 - 661
  8. Find a number that leaves a remainder of 2 when divided by 3, a remainder of 3 when divided by 4, and a remainder of 4 when divided by 5. What is the smallest such number? Can you give a simple explanation of why it is the smallest?

A Shortcut for Divisibility by

  1. Find, without dividing, whether the following numbers are divisible by 9.
    • (i) 123
    • (ii) 405
    • (iii) 8888
    • (iv) 93547
    • (v) 358095
  2. Find the smallest multiple of 9 with no odd digits.
  3. Find the multiple of 9 that is closest to the number 6000.
  4. How many multiples of 9 are there between the numbers 4300 and 4400?

Digital Roots

  1. The digital root of an 8-digit number is 5. What will be the digital root of 10 more than that number?
  2. Write any number. Generate a sequence of numbers by repeatedly adding 11. What would be the digital roots of this sequence of numbers? Share your observations.
  3. What will be the digital root of the number 9a + 36b + 13?
  4. Make conjectures by examining if there are any patterns or relations between
    • (i) the parity of a number and its digital root.
    • (ii) the digital root of a number and the remainder obtained when the number is divided by 3 or 9.

Digits in Disguise

  1. If 31z5 is a multiple of 9, where z is a digit, what is the value of z? Explain why there are two answers to this problem.
  2. "I take a number that leaves a remainder of 8 when divided by 12. I take another number which is 4 short of a multiple of 12. Their sum will always be a multiple of 8", claims Snehal. Examine his claim and justify your conclusion.
  3. When is the sum of two multiples of 3, a multiple of 6 and when is it not? Explain the different possible cases, and generalise the pattern.
  4. Sreelatha says, "I have a number that is divisible by 9. If I reverse its digits, it will still be divisible by 9".
    • (i) Examine if her conjecture is true for any multiple of 9.
    • (ii) Are any other digit shuffles possible such that the number formed is still a multiple of 9?
  5. If 48a23b is a multiple of 18, list all possible pairs of values for a and b.
  6. If 3p7q8 is divisible by 44, list all possible pairs of values for p and q.
  7. Find three consecutive numbers such that the first number is a multiple of 2, the second number is a multiple of 3, and the third number is a multiple of 4. Are there more such numbers? How often do they occur?
  8. Write five multiples of 36 between 45,000 and 47,000. Share your approach with the class.
  9. The middle number in the sequence of 5 consecutive even numbers is 5p. Express the other four numbers in sequence in terms of p.
  10. Write a 6-digit number that is divisible by 15, such that when the digits are reversed, it is divisible by 6.
  11. Deepak claims, "There are some multiples of 11 which, when doubled, are still multiples of 11. But other multiples of 11 don't remain multiples of 11 when doubled". Examine if his conjecture is true; explain your conclusion.
  12. Determine whether the statements below are 'Always True', 'Sometimes True', or 'Never True'. Explain your reasoning.
    • (i) The product of a multiple of 6 and a multiple of 3 is a multiple of 9.
    • (ii) The sum of three consecutive even numbers will be divisible by 6.
    • (iii) If abcdef is a multiple of 6, then badcef will be a multiple of 6.
    • (iv) 8(7b - 3) - 4(11b + 1) is a multiple of 12.
  13. Choose any 3 numbers. When is their sum divisible by 3? Explore all possible cases and generalise.
  14. Is the product of two consecutive integers always a multiple of 2? Why? What about the product of three consecutive integers? Is it always a multiple of 6? Why or why not? What can you say about the product of 4 consecutive integers? What about the product of five consecutive integers?
  15. Solve the cryptarithms —
    • (i) EF × E = GGG
    • (ii) WOW × 5 = MEOW
  16. Which of the following Venn diagrams captures the relationship between the multiples of 4, 8, and 32?
Sample

Worked examples

3 questions from Number Play, solved step by step.

What Remains?

The sum of four consecutive numbers is 34. What are these numbers?

Let the four consecutive numbers be n, n+1, n+2, n+3.

Their sum: n + (n+1) + (n+2) + (n+3) = 4n + 6 = 34

4n = 34 - 6 = 28

n = 7

The four consecutive numbers are 7, 8, 9, 10.

Verification: 7 + 8 + 9 + 10 = 34 ✓

Answer: 7, 8, 9, and 10

A Shortcut for Divisibility by

Find, without dividing, whether the following numbers are divisible by 9. (i) 123 (ii) 405 (iii) 8888 (iv) 93547 (v) 358095

A number is divisible by 9 if and only if the sum of its digits is divisible by 9.

(i) 123: digit sum = 1 + 2 + 3 = 6. 6 is not divisible by 9. Not divisible by 9.

(ii) 405: digit sum = 4 + 0 + 5 = 9. 9 is divisible by 9. Divisible by 9. ✓

(iii) 8888: digit sum = 8 + 8 + 8 + 8 = 32. 32 is not divisible by 9. Not divisible by 9.

(iv) 93547: digit sum = 9 + 3 + 5 + 4 + 7 = 28. 28 is not divisible by 9. Not divisible by 9.

(v) 358095: digit sum = 3 + 5 + 8 + 0 + 9 + 5 = 30. 30 is not divisible by 9. Not divisible by 9.

Answer: Only 405 (ii) is divisible by 9

Digital Roots

The digital root of an 8-digit number is 5. What will be the digital root of 10 more than that number?

The digital root of a number equals its remainder when divided by 9 (with digital root 9 when the remainder is 0).

If digital root = 5, then the number ≡ 5 (mod 9).

Adding 10: the new number ≡ 5 + 10 = 15 ≡ 6 (mod 9).

So the digital root of the new number is 6.

Example: 40000001 has digit sum 4+0+0+0+0+0+0+1 = 5 (digital root 5).

40000001 + 10 = 40000011; digit sum = 4+0+0+0+0+0+1+1 = 6 (digital root 6). ✓

Answer: 6

All 32 questions in Number Play are solved step by step in the Solutions PDF above.

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More solutions in Ganita Prakash

01A Square and a Cube02Power Play03A Story of Numbers04Quadrilaterals06We Distribute, Yet Things Multiply07Proportional Reasoning-108Fractions in Disguise09The Baudhayana-Pythagoras Theorem10Proportional Reasoning-211Exploring Some Geometric Themes12Tales by Dots and Lines13Algebra Play14Area
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More NCERT Solutions for Class 8

Read the Number Play textbook chapter / PDF, or browse all NCERT Class 8 Mathematics solutions.

← PreviousCh 4: QuadrilateralsNext →Ch 6: We Distribute, Yet Things Multiply

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