Chapter 5 Number Play (Class 8 - Latest Maths NCERT (Ganita Prakash I) Solutions)

Question 1. The sum of four consecutive numbers is $34$. What are these numbers?

Question 2. Suppose $p$ is the greatest of five consecutive numbers. Describe the other four numbers in terms of $p$.

Question 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 divisible by $6$.

(iv) The sum of a multiple of $6$ and a multiple of $9$ is a multiple of $3$.

(v) The sum of a multiple of $6$ and a multiple of $3$ is a multiple of $9$.

Question 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.

Question 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?”

Question 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?

Question 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) $4779 - 661$

Question 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?

Question 1. Find, without dividing, whether the following numbers are divisible by $9$.

(i) $123$

(ii) $405$

(iii) $8888$

(iv) $93547$

(v) $358095$

Question 2. Find the smallest multiple of $9$ with no odd digits.

Question 3. Find the multiple of $9$ that is closest to the number $6000$.

Question 4. How many multiples of $9$ are there between the numbers $4300$ and $4400$?

Question 1. The digital root of an $8$-digit number is $5$. What will be the digital root of $10$ more than that number?

Question 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.

Question 3. What will be the digital root of the number $9a + 36b + 13$?

Question 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 $9$.

Question 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.

Question 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.

Question 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.

Question 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 $9$?

Question 5. If $48a23b$ is a multiple of $18$, list all possible pairs of values for $a$ and $b$.

Question 6. If $3p7q8$ is divisible by $44$, list all possible pairs of values for $p$ and $q$.

Question 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?

Question 8. Write five multiples of $36$ between $45,000$ and $47,000$.

Share your approach with the class.

Question 9. The middle number in the sequence of $5$ consecutive even numbers is $5p$. Express the other four numbers in sequence in terms of $p$.

Question 10. Write a $6$-digit number that it is divisible by $15$, such that when the digits are reversed, it is divisible by $6$.

Question 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.

Question 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 $12$.

Question 13. Choose any $3$ numbers. When is their sum divisible by $3$? Explore all possible cases and generalise.

Question 14. Is the product of two consecutive integers always multiple of $2$? Why? What about the product of these 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?

Question 15. Solve the cryptarithms —

(i) $EF \times E = GGG$

(ii) $WOW \times 5 = MEOW$

Question 16. Which of the following Venn diagrams captures the relationship between the multiples of $4$, $8$, and $32$?