CBSE Class 6 Maths Playing with Numbers HOTs

HOTS

1. 105 goats, 140 donkeys and 175 cows have to be taken across a river. There is only one boat which will have to make many trips in order to do so. The lazy boatman has his own conditions for transporting them. He insists that he will take the same number of animals in each trip and they have to be of the same kind. He will naturally like to take the largest possible number each time. Can you tell how many animals will go in each trip?
Answer: 35

2. Sergeant Band was taking part in a parade and it was necessary to have equal number of soldiers in each row while marching. When he tried to arrange them in rows of two, he had one soldier left. The same thing happened when he tried to arrange them in rows of three, four, six or eight. What is the least number of soldiers he could have had? Could he have arranged his band so that every row had an identical number of soldiers?
Answer: 25 rows of 5

3. An athlete jogs everyday, swims every other day, lifts weights every 3rd day. Therefore, he swims, jogs and lifts every 6th day. In addition he plays tennis on Sundays. After how many days will he do all the activities on the same day?
Answer: 42

4. Determine two numbers nearest to 10,000 which are exactly divisible by 3, 4, 5 and 6.
Answer: 10020; 9960

5. A boy saves ` 4.65 daily. Find the least number of days in which he will be able to save an exact number of rupees.
Answer: 20 days

CHALLENGES

1. Find the factors of 360.

2. Find the prime factorisation of 360.

3. Find the largest value of a such that 3aa3 is divisible by 77.

4. Apalindrome is a number which reads the same from left to right and right to left. For example 13531 is a 5-digit palindrome. Show that any 4-digit palindrome is divisible by 11.

5. Find the largest 5-digit palindrome divisible by 11.

6. You can develop a divisibility test by 7 as follows. Take a number, say 94976. Make groups of 3 digits starting from right and the last group may contain 1, 2 or 3 digits. Put alternatively + and – signs and compute the absolute value. If this is divisible by 7, then the original number is divisible by 7. Here the groups are 94 and 976. We compute 976 – 94 = 882.
For 882, you may apply the test you have learnt earlier;
882 → 88 – (2 × 2) = 88 – 4 = 84;
→ 8 – (2 × 4) = 8 – 8 = 0.
This shows that 94976 is divisible by 7.
Apply this method to the numbers 5527578, 608150, 552907378, and 5529073767 to test their divisibility by 7.
(The advantage of this method is the number of digits reduces fast when the given number is very large. This test also applies for 11 and 13.)

7. Consider all 4-digit numbers which can be formed using the digits 2,3,4,5, each exactly once.
a. How many numbers can you form?
b. How many of these are divisible by 4?
c. How many of these are divisible by 3?
d. How many of these are divisible by 5?
e. How many of these are divisible by 11?

8. Consider the set {1, 3, 4, 5, 6, . . .}, obtained by removing 2 from the system of natural numbers.
List the first 5 prime numbers in this system. (A number in this system is a prime only if it has exactly two distinct factors.)

9. Consider the set {1, 5, 9, 13, . . .}; all those natural numbers which leave remainder 1 when divided by 4. Prove that the product of any two numbers in this set is again a number in this set.
(We say the set is closed under multiplication.) Find the first 6 prime numbers in this collection.

10. Consider 13 and 23. Both are prime numbers and they differ by 10. Find few more examples of pairs of prime numbers which differ by 10. Find at least two examples of pairs of prime numbers which differ by 100.

11. Consider 13 and 31. Both are prime numbers and one is obtained by reversing the digits of the other. Find few more examples of such pairs. Find an example of a 3-digit prime number such that any shuffling of the digits again gives a prime number.

12. Consider the numbers 90, 91, 92, 93, 94, 95, 96. These are 7 consecutive numbers, all being composite numbers. Can you give an example of 8 consecutive numbers all of them being composite. Can you make it 9? 10? Any given number?
(Hint: Consider product of consecutive numbers from 1 to a given number. For example 1×2×3×·
· ·×10 is called ten factorial and written as 10!. Such numbers are useful in constructing any arbitrarily large collection of consecutive numbers which are all composite.)

13. Exploration: Here are some small perfect numbers: 6, 28, 496, 8128, 3350336. The number of digits grow fast. So far only 51 perfect numbers are known and the largest known perfect number has 4,97,24,095 digits in it!. All the perfect numbers known are even. No one knows whether odd perfect number exists. It is also not known whether there are infinitely many perfect numbers. Learn more about this by going to literature.

14. Find the smallest number which leave the same non-zero remainder when divided by 2,3,5, and which is completely divisible by 7.

15. Find the smallest number which leave the same non-zero remainder when divided by 2,3,5,7 and which is completely divisible by 11.

16. Find the smallest number which leave the remainders 1,2,3 when divided respectively by 5,6,7.
(Hint: First make a list of numbers which leave remainder 1 when divided by 5. Make another list of numbers which leave remainder 2 when divided by 6. Find the first common element in these two lists. If you add any multiple of LCM of 5, 6 to this number, that gives a number which leave remainders 2, 3 when divided respectively by 5, 6. Go on adding the multiples of LCM of 5,6 till you get a number which leaves remainder 3 when divided by 7).

17. Can you have two distinct numbers such that their HCF and LCM are the same?

18. You know that the product of two numbers is the same as the product of their HCF and LCM.
For which numbers it can happen that the sum of two numbers is the same as the sum of their HCF and LCM?

19. Find two numbers a, b such that HCF(a, b) × HCF(a, b) = LCM(a, b). Construct more examples of such pairs.

SUMMARY

1. When two or more numbers are multiplied to give a product, then each of the numbers is called a factor of the product. Also the product is called a multiple of that number.

2. 1 is a factor of every number and 0 is a multiple of every number.

3. Every non-zero number is a factor as well as a multiple of itself.

4. Numbers having exactly two factors namely 1 and the number itself, are called prime numbers.

5. Numbers having more that two factors are called composite numbers.

6. 1 is neither prime nor composite.

7. The only even prime number is 2.

8. When a number is written as the product of the prime factors it is called prime factorisation.

9. HCF of two or more numbers is the highest factor that divides the number exactly without leaving a remainder.

10. LCM of two or more numbers is the least number that gets divided by the given numbers.

11. LCM×HCF = Product of numbers

ERRORANALYSIS

1. Instead of finding the HCF, students find the LCM and vice versa.

2. Students fail to interpret the question.

ACTIVITY I

In an alarm clock/mobile phone there is an option 'Snooze Duration'. It is the duration of repeat alarm until it is stopped.
Take four different types of alarm clocks/mobile phones.
Set the alarm of these clocks for the same time. Let the time be 6 am. 

The time set in all the alarm clocks is 6 am. At what time after 6 am, all the alarms will ring again?
The time at which they all rang together is the _______________ of the snooze duration.

ACTIVITY II

Objective: To explore the concept of HCF of two numbers.
Requirements: Geometry box, ruler, a pair of scissors, paper sheet.

Steps:
Choose any two numbers whose HCF is to be found out. Say, 24 and 36.
1. Cut a rectangle having its length 36 cm and breadth 24 cm. 

2. Cut out the largest possible square from it. The other plane figure we get is a rectangle of dimensions 24 cm × 12 cm. 

3. Repeat the above step to cut out the largest possible square from the rectangle formed above. 

4. Continue the activity till all the plane figures formed are squares.

Conclusion :
We see that two squares of sides 12 cm are obtained by cutting the rectangle with dimensions 36cm × 24cm and there is no other rectangle left. Thus, the HCF of 24 and 36 is 12.
We can find the HCF of the other given numbers by using this activity.

ACTIVITY III

Puzzle

1. Each of the 9 squares in Table (a) shown below has a number and corresponding alphabet that you should use to find the name of a popular animation film. To find the name of the film, find the HCFs of the rows shown below in Tables (b), (c), and (d) and look up the number in Table (a) for the corresponding letter.