Number Play Class 8 Worksheet with Answers Maths Chapter 5

Complete the Ganita Prakash Class 8 Worksheet and NCERT Class 8 Maths Chapter 5 Number Play Worksheet with Answers before your unit tests for better preparation.

Number Play Worksheet Class 8

Class 8 Maths Number Play Worksheet

Number Play Class 8 Ganita Prakash Worksheet

Sum of Consecutive Numbers

Question 1.
Ravi arid Meera are helplng their uncle in arranging chairs in rows for a community event.

Ravi: “Uncle says there are 15 chairs. If we arrange them in 3 rows, we will have 5 in each row.”

Meera: “But we can also arrange them in 5 rows of 3. That’s interesting!”

Uncle: “Yes, but did you know 15 can also be written as the sum of consecutive numbers?”

Ravi: “Consecutive numbers? Like 7 + 8?”

Uncle: “Exactly! 15 = 7 + 8. And also 15 = 4 + 5 + 6.”

Meera (excited): “Oh! So some numbers can be written in many ways as sums of consecutive numbers.

(a) Can you find another set of consecutive numbers that also adds to 15? Write them.
Answer:
Yes,1 + 2 + 3 + 4 + 5 = l5

(b) Write 21 as a sum of consecutive numbers in at least two different ways.
Answer:
10 + 11 = 21,6 + 7 + 8 = 21

(c) Meera said, “Some numbers can be written in many ways.” Check if 9 can be written as the sum of consecutive numbers and in how many other ways?
Answer:
Yes, 4 + 5 = 9, 2 + 3 + 49

(d) Can all odd numbers be written as the sum of consecutive numbers? Test this using 11, 13, and 17. Write what you notice.
Answer:
No,8(Answer may vary)

(e) ‘Can every number be written as a sum of consecutive numbers?’ Give one example of a number that cannot be expressed this way.
Answer:

Question 2.
Fill in the Blanks

(a) The number 36 can be written as the sum of consecutive numbers in following ways. (Find at least two)
Answer:
11 + 12 + 13 = 36, 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36

(b) 12 = _____ + _____ + (Write it as sum of three consecutive numbers).
Answer:
3 + 4 + 5 = 12

Question 3.
Write 45 as:
(a) a sum of 2 consecutive numbers
(b) a sum of 3 consecutive numbers
(c) a sum of 5 consecutive numbers
Answer:
(a) 22 + 23 = 45
(b) 14 + 15 + 16 = 45
(c) 7 + 8 + 9 + 10 + 11 = 145

Question 4.
Which of these numbers can be written as a sum of consecutive numbers: 10, 16, 25, 36?
Answer:
10, 25 and 36

Question 5.
Find a number between 50 and 60 that can be written as a sum of consecutive numbers in at least 3 different ways.
Answer:
55 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
55 = 9 + 1 0 + 11 + 12 + 13
55 = 27 + 28 (Answers may vary)

Question 6.
Check whether 100 can be expressed as a sum of consecutive numbers. If yes, give one such expression.
Answer:
Yes, 100 = 18 + 19 + 20 + 21 + 22

Question 7.
Write the general rule – ‘When can a number be written as the sum of two consecutive natural numbers?’
Answer:
(n)+(n + 1) = 2n + 1

Question 8.
Put ‘+’ or ‘-‘ between 11, 12, 13, 14, and calculate at least 4 different results.

Question 9.
Can changing ‘+’ or ‘-‘ between even numbers turn an even sum into an odd result?
Answer:
No

Question 10.
Without calculating, predict whether the result of 101 – 103 – 105 – 107 will be odd or even.
Answer:
Even

Question 11.
Is it possible to insert ‘+’ and ‘-’ signs between four consecutive numbers so that the result is always divisible by 4? Try and explain.
Answer:
Yes, 5 + 6 – 7 = 4; 5 + 6 – 7 + 8 = 12 (Answer may vary)

Breaking Even

Question 12.
Aarav and Nisha are playing a number game with cards. Each card has a number written on it.

Aarav: Look, I have 3 and 5. When I add them, 3 + 5 = 8, which is even.

NisKa: I will try 6 and 4. Their sum is 10, also even.

Aarav (thinking): Wait… adding two odd numbers gives even. Adding two even numbers also gives even!

Niska: But what if we add one odd and one even?

Aarav: Let us try 5 + 6 = 11. Oh! That’s odd.

Nisha (excited): So there’s a pattern in odd and even numbers!

Based on the above, answer the following questions:

(a) If Aarav added 8 and 11. What did he get? Was it odd or even?
Answer:
19, odd

(b) If Nisha added 12 and 16. What was the result? Was it odd or even?
Answer:
28, even

(c) Complete the rule for the following:
(i) Odd + Odd =
(ii) Even + Even
(iii) Odd + Even =
(iv) Odd – Odd =
(V) Odd – Even =
(vi) Even – Even =
Answer:
(i) Even
(ii) Even
(iii) Odd
(iv) Even
(v) Odd
(vi) Even

(d) What will be the sum of 101 + 205? Justify using the rule.
Answer:
306

Question 13.
State True or False for the following statements:
(a) The sum of two even numbers is always odd.
Answer:
False

(b) The sum of an odd and an even number is always odd.
Answer:
True

Question 14.
If n is even, what can you say about n² + 3n? (Odd or even?)
Answer:
Even

Question 15.
If n is odd, what can you say about n² – n?
Answer:
Even

Pairs to Make Fours

Question 16.
Manvi and Aarush are playing a board game with tokens. Each token has a number on it.

Manvi: “I’ve got tokens with numbers 2, b, 10, 14 … Whenever I divide them by 4, the remainder is always 2.”

Aarush: “And I’ve got tokens with numbers 4, 8, 12, 1 b … They’re all exact multiples of 4.

Manvi (excited): “Wait! If I pick one of my tokens and add it to another one of mine, the sum is always a multiple of 4!”
Aarush: “Let’s test it, b + 10 = 1b, which is divisible by 4!”

Manvi: “Yes! Any two numbers that each leave a remainder of 2 when divided by 4 will add up to a multiple of 4. That’s how we make pairs to make 4!”

Based on the above, answer the following Questions.
(a) Give two numbers of the form (4k) and (4k + 2). Show that their sum is never divisible by 4.
Answer:
8 + 10 = 18 not divisible by 4; 12 + 14= 26 not divisible by 4 (Answers may vary)

(b) Can two odd numbers ever form a “pair to make four”? Explain with an example.
Answer:
Yes

(c) Show that if a number is of the form 4k and another is of the form 4m, then their sum is also divisible by 4.
Answer:
Do it yourself

(d) Show that if a number is of the form 4k + 2 and another is of the form 4m + 2, then their sum is divisible by 4.
Answer:
Do it yourself

Question 17.
Answer the following questions:

(a) Write the general form of numbers leaving remainder 1 and remainder 3 when divided by 4, but their sum is multiple of 4.
Answer:
4k + 1, 4k + 3

(b) Prove that if both numbers leave a remainder of 1 when divided by 4, their sum leaves a remainder of 2 when divided by 4.
Answer:
Do it yourself

Question 18.
If a number is divisible by both 7 and 6, it must be divisible by 42. Examine whether it is ‘Always true’, ‘Sometimes true’ or ‘Never true’.
Answer:
Always True

Question 19.
Examine, whether it is “Always true”, ‘Sometimes ture’, or ‘Never true’, that if a number is divisible by 3 and another number is divisible by 4, then their sum is divisible by 7.
Answer:
Sometimes

What Remains?

Riya was playing with numbers and tried the expression 5k – 2.
For k = 1,2, 3, 4, 5, she got 3, 8, 13, 18, 23, …
She noticed each was 3 more than a multiple of 5.
So she realized such numbers can also be written as 5k + 3

Question 20.
Suppose q is the smallest of six consecutive numbers. Write the other five numbers in terms of q.
Answer:
q, q + 1, q + 2, q + 3, q + 4, q + 5

Question 21.
If a number leaves a remainder 4 when divided by 9, express such numbers in algebraic form. Find the smallest three such numbers greater than 50.
Answer:
9k + 4; 58, 67, 76

Question 22.
A bus arrives at a station every 12 minutes and a train every 18 minutes. If both arrive together at 9:00 a.m., at what time will they arrive together again? Express this problem in terms of remainders or multiples.
Answer:
9 ; 36 am

Question 23.
When divided by 7, the number 215 leaves a remainder 5, and the number 348 leaves a remainder 6. Without calculating, find the remainders when these expressions are divided by 7:
(a) 215 + 348
(b) 348 – 215
Answer:
(a) 3
(b) 0

Checking Divisibility Quickly

Question 24.
Without dividing, check which of the following are divisible by 10: 7040, 145, 8800, 91450.
Answer:
7040, 8800, 91450

Question 25.
Check those divisible by 5 or 10 or by both: 8450, 110, 250, 6795, 10081, 55650.
Answer:
Divisible by 5: 8450, 110, 250, 6795, 55650
Divisible by 10: 8450, 110, 250, 55650
Divisible by both: 8450, 110, 250, 55650

Question 26.
Check which are divisible by 2: 184, 359, 222, 707, 6008, 10111?
Answer:
184, 222, 6008

Question 27.
A 5-digit number is written as edcba = e × 10⁴ + e × 10³ + c × 10² + b × 10 + a. Use this to justify the shortcuts for checking divisibility by 10, 5 and 2.

Question 28.
(a) State and explain the shortcut for checking divisibility by 4 using place value. Check: 516, 732, 9052, 1201 are divisible by 4 or not.
Answer:
516, 732 are divisible by 4

(b) Fill one digit to make the number divisible by 4: 78
Answer:
4 (Answer may vary)

Question 29.
(a) State and explain the shortcut for checking divisibility by 8 using place value. Check: 104, 1304, 5216, 624, 1016 are divisible by 8 or not.
Answer:
104, 1304, 5216, 624, 1016 are divisible by 8

(b) Fill one digit to make the number divisible by 8 (look at the last three digits): 72 – 0.
Answer:
0 (Answer may vary)

Question 30.
Without dividing, determine whether each of these is divisible by 9. (Use the sum of digits test)
(a) 256
(b) 729
(c) 11112
(d) 80721
Answer:
729 and 80721 are divisible by 9

Question 31.
(a) Find the multiple of 9 closest to 8200.
Answer:
8199

(b) How many multiples of 9 are there between 2380 and 2440?
Answer:
7 (Seven)

Question 32.
Test divisibility the following numbers for by 3:
(a) 2547
(b) 8068
(c) 80721
(d) 100055
Answer:
2547 and 8072 are divisible by 3

Question 33.
Using the shortcut method, determine whether the following numbers are divisible by 11. If a number is not divisible, state its remainder.
(a) 2953
(b) 50318
(c) 76219
(d) 110891
Answer:
(o) Not divisible, 5
(b) Not divisible, 4
(c) Divisible
(d) Divisible

Question 34.
(a) Compute the alternating sum of digits for 462 and decide divisibility by 11.
(b) Insert suitable digits in the blanks of 5_46_ so that the number becomes divisible by 11. Show the steps of your calculations.
Answer:
(a) 0, Divisible by 11
(b) 50468 (Answer may vary)

Digital Roots

Question 35.
Find the digital roots of the following numbers:
(a) 50876
(b) 99999
(c) 202581
(d) 40003
Answer:
(a) 8
(b) 9
(c) 9
(d) 7

Question 36.
Without full addition, predict the digital root of 123456 + 654321. Explain.
Answer:
6

Question 37.
(a) Between 240 and 300, which numbers have a digital root 1?
(b) Between 950 and 1000, list the numbers with a digital root 7.
Answer:
(a) 244, 253, 262, 271, 280, 289, 298
(b) 952, 961, 970, 979, 988, 997

Question 38.

(a) Write the digital roots of 15 consecutive numbers starting from 342. What pattern you observe?
Answer:
342 → 9; 343 → 1; 344 → 2; 345 → 3; 346 → 4; 347 → 5; 348 → 6, 349 → 7, 350 → 8; 351 → 9; 352 → 1; 353 → 2; 354 → 3; 355 → 4; 356 → 5 (Digital roots repeat in a cycle of 9)

(b) Start at 9995 and write the digital roots of the next 12 numbers. What happens when you “cross” 10,000?
Answer:
Do it yourself

(c) Find the digital roots of the first ten multiples of 7. What cyclic pattern do you see?
Answer:
7 → 7, 14 → 5, 21 → 3, 28 → 1, 35 → 8, 42 → 6, 49 → 4, 46 → 2, 63 → 9, 70 → 7 (The digital roots repeat in a cycle of 9)

(d) What will be the digital root of number 7x + 21 y + 3?
Answer:
7 + x + 2 + 1 + y + 3 = 13 + x + y = 4x + y

Question 39.
Fill one digit so the digital root is 7: 5_283.
Answer:
57283

Digits in Disguise (Cryptarithms)

Question 40.
If A + A = B and B is a two-digit number like 22, 44, 66, …, what values can A and B take?
Answer:
A = 11, 22, 33, 44,….
B = 22, 44, 66, 88, …

Question 41.
In the puzzle SEND + MORE = MONEY, how many letters are there? Do you think each letter stands for a unique digit? Identify the digits.
Answer:
8; S → 9, E → 5, N → 6, D → 7, M → 1, 0 → 0, R → 8, y → 2

Question 42.
If T + T = F in the TWO + TWO = FOUR puzzle, what possible digits can F be?
Answer:
2,4,6, 8

Question 43.
Solve the following:
(a) \(\begin{array}{r}
5 \mathrm{~A} \\
+5 \mathrm{~A} \\
\hline 11 \mathrm{~B}
\end{array}\)

(b) \(\begin{array}{r}
\text { CAT } \\
+ \text { CAT } \\
\hline \text { DOG }
\end{array}\)

(d) UV x 6 = WUV
(e) JK x 3 = LLL
(f) CD x 5 = ECD
Answer:
(a) A = 5, B = O
(b) CAT → 217 D0G → 434
(c) B → 0
(d) UV → 18, WUV → 108
(e) JK → 37; LLL → 111
(f) CD → 25, ECD → 125 (Answer may vary)

Question 44.
Prove that for any fixed integers a, b, c, d, alt eight expressions obtained by placing ‘+’ and ‘-‘ signs in a ± b ± c ± d have the same parity. Use either a sign-switch argument or parity rules.
Answer:
Do it yourself

Question 45.
If 4y23 is a multiple of 9, where.y is a single digit, find the value(s) of y. Explain why there are two possible answers to this question.
Answer:
0 and 9

Question 46.
If 5c17d is a multiple of 18, list all possible pairs of values for c and d.
Answer:
(5, 0), (3, 2), (1, 4), (8, 6), (6, 8)

Worksheet On Number Play Class 8

A. Choose the correct option.

1. Without dividing, identify which of the following numbers is divisible by q?
(a) 6532
(b) 9909
(C) 42031
(d) 75214
Answer:
(b) 9909

2. Which of the following expressions is always even for any integers m and n?
(a) 2m + 3n
(b) 4m + 2n
(c) m2 + 1
(d) 3m + 5n
Answer:
(b) 4m + 2n

3. A number is divisible by 24 if and only if it is divisible by:
(a) 4 and 6
(b) 12 and 2
(c) 3 and 8
(d) 9 and 4
Answer:
(c) 3 and 8

4. The digital root of p87654 is:
(a)3
(b)6
(c)9
(d)1
Answer:
(a)3

5. Which of the following is divisible by 11?
(a) 158
(b) 481
(c) 5529
(d) 90904
Answer:
(d) 90904

Directions. (For Q.6 – 7): In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option as:
(a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false, but Reason (R) is true.

6. Assertion (A): For any integers a, b, c, d, all expressions of the form a ± b ± c ± d have the same parity.
Reason (R): Changing any one V or sign in such an expression changes its value by an even number.
Answer:
(a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

7. Assertion (A): If a number is divisible by 24, it is divisible by 4 and 6 both.
Reason (R): If a number is divisible by k, then it is divisible by all the factors of k.
Answer:
(a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

B. Fill in the blanks.

1. Numbers that leave a remainder of 3 when divided by 5 can be written as _______
Answer:
5K + 3

2. If the sum of the digits of a number N is 27, then the digital root of N is _______
Answer:
9

3. The alternating sum-of-digits method checks divisibility by _______.
Answer:
11

4. The sum of two even numbers not multiples of 4, is always a multiple of _______.
Answer:
4

5. If a number is divisible by 9, then it is necessarily divisible by _______.
Answer:
3

C. State whether the following statements are True (T) or False (F).

1. If a number is divisible by 12, then it is divisible by all the factors of 12.
Answer:
True

2. If a number is divisible by both 9 and 4, it must be divisible by 36.
Answer:
True

3. The sum of an odd number and an even number is always odd; therefore, it can never be a multiple of 6.
Answer:
True

4. Among the eight expressions formed by placing V and between four consecutive numbers, every value is even.
Answer:
True

5. If the sum of the digits of a number is not divisible by 9, then the number is not divisible by 9.
Answer:
True

D. Solve the following.

Question 1.
Without using long division, find the remainders when CO 93547 and GO 358095 are divided by 9. State whether each is divisible by 9.
Answer:
(i) 1, Not divisible
(ii) 3 Not divisible

Question 2.
Using the alternating-sum-of-digits method, determine whether 5529 and 857076 are divisible by 11. If not, state the remainder in each case.
Answer:
5529 → Not divisible, Reminder → 7, 857O76 → divisible

Question 3.
Find the smallest positive integer that leaves a remainder 3 when divided by 5 and a remainder 2 when divided by 3.
Answer:
8

Question 4.
Solve the cryptarithm with distinct dtgits: XY × 4 = ZX (X ≠ O). List all possible solutions, if any.
Answer:
xy → 23, zx → 92

Question 5.
Explain why checking divisibility by 4 and by 6 is not sufficient to conclude divisibility by 24, but checking divisibility by 8 and by 3 is sufficient. Justify using prime factorisation.
Answer:
Do it yourself

Question 6.
Tathagat claims: “If you add any three numbers that each leave a remainder of 2 when divided by 6, the sum is always a multiple of 6.” Investigate, give examples and counter-examples if needed, and generalise.
Answer:
Claim is true (6K₁ + 2) + (6k₂ + 2) + (6k₃ + 2) = 6(K₁ + K₂ + K₃ + 1)

Question 7.
Start with any whole number and generate a sequence by repeatedly adding 11. Describe the pattern of digital roots you observe. What is the cycle length and why?
Answer:
Do it yourself

Question 8.
Write an algebraic expression for all numbers that are 4 less than a multiple of 7 and 1 more than a multiple of 3. Find such least positive number and explain your reasoning.
Answer:
7K – 4 and 3K + 1, 10 (Cycle length = 9)