CBSE Class 8 Maths Playing with Number HOTs

HOTS

1. Fill the grid so that each row, column and 3x3 square box contains the digits 1 to 9, without repeating the number in the same row, column or box. (You cannot change the number already given in the grid.) 

Answer:

2. In the following problem, replace the letters of English alphabet by digits (two or more letters may have the same value) to complete the procedure of division. 

Answer: A=8 E=5
B=6 F=6
C=4 G=3
D=4 H=6

3. In the 4 × 5 array given below, place the numbers from 1 to 20 (without repetition) so that a sequence (chain of numbers) is obtained from 1 to 20. You can move to the adjacent cell to the left, right, up or down but not diagonally. The positions of the numbers that already exist are fixed. 

Answer: 

4. Fill in the circle with the correct digits. 

Answer:

CHALLENGES

1. Is 12345 divisible by 15?

2. Is 124956 divisible by 36?

3. Prove that the number x_k x_k–1 · · · x₁x₀ is divisible by 11, if and only if (x_k+x_k–2+· · · ) – (x_k–1+x_k–3+· · · ) is divisible by 11.

4. Prove that 5746312 is divisible by 88.

5. What is the smallest 5-digit number divisible by 11 and which contains each of the digits 2, 3, 4, 5, 6?

6. Using the digits 4, 6, 7, 8, 9, each once, construct the smallest 5-digit number divisible by 44.

7. How many 5-digit numbers are formed by the digits 2, 3, 4, 5, 6 each used once, which are divisible by 11? Of these, how many are divisible by 4?

8. Find the smallest natural number that is divisible by 5 and the sum of its digits is 99.

9. What is the smallest value of 'a' such that, when you write a, a+1, a+2 in a row, the number is divisible by 11.

10. Suppose a number with more than 1 digit is divisible by 11. Is the number obtained by reversing all the digits also divisible by 11? Give reasons.

11. Suppose the number 4a3b is divisible by 11. Find all possible values of a and b.

12. How many 3-digit numbers are there with one of the digits as 7 and the other two digits are distinct digits such that each number is divisible by all its digits?

13. Find all 4-digit numbers with 5 as one of the digits and others distinct digits, such that the number is divisible by each of its digits?

14. Exploration : How many 4-digit numbers are there
a. with distinct digits
b. with not necessarily distinct digits
such that each number is divisible by all its digits?

15. The following alphamatics have no solution. Explain why? 

16. Find all the digits in the given sums (There may be more than one solution.) 

17. Solve the following Alphamatics. 

18. Find the missing digits in the long division below: 

19. Find all the missing digits in the long division below : 

20. In the given division, exactly one of the digits is wrong. Find it, correct it and find all the missing digits. 

21. Find all the missing digits in the following multiplication. 

SUMMARY

1. In general form, a 2-digit number ab can be written as 10a+b whereas a 3-digit number abc can be written as 100a+10b+c.

2. Sum of a 2-digit number and the number formed by reversing its digits is always divisible by 11 and the sum of the digits.

3. Difference of a 2-digt number and the number formed by reversing its digits is divisible by 9 and the difference of the digits.

4. The sum of a 3-digit number abc and the numbers obtained by changing the order of the digits cyclically (i.e. bca and cab) is divisible by 111, 37, 3 and (a+b+c).

5. The difference of a 3-digit number abc and the number obtained by reversing the digits is divisible by 99 and the difference of the digits at the hundreds and units place.

6. A number is divisible by 2 if its unit digit is 0,2,4,6 or 8.

7. A number is divisible by 3 if the sum of the digits is divisible by 3.

8. A number is divisible by 4 if the number formed by the last two digits is divisible by 4.

9. A number is divisible by 5 if the units digit is 0 or 5.

10. A number is divisible by 6 if it is divisible by both 2 and 3.

11. A number is divisible by 8 if the number formed by the last three digits is divisible by 8.

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

13. A number is divisible by 11 if the difference of the sum of digits at the odd places and the sum of digits at the even places is either 0 or divisible by 11.

14. A number is divisible by 11 if the sum of digits in blocks of two digits from right to left is divisible by 11.

ERROR ANALYSIS

1. Students tend to make errors while writing numbers in the general form.

2. In questions where units digit, tens digit and hundreds digit are given in some ratio, students tend to take the digits in the wrong ratio.

3. Errors are made while applying divisibility tests specially in case of divisibility rules of 7 and 11.

ACTIVITY

Objective : To decode the message
Step 1 : Work in pairs and solve the sum given in the box with the letter Q.
Step 2 : The answer will match with a number given in a circle in another box. Look for it.
Step 3 : Copy the letter given in that box in the first blank of the message.
Step 4 : Solve the sum given in that box.
Step 5 : Repeat steps 2, 3 and 4 till you reach the first box.
Step 6 : Read the message

THE EARTH IS ROUND.