NCERT Solutions Class 6 Math Ganita Prakash Chapter 5 Prime Time
Get NCERT Class 6 Math Ganita Prakash 5th chapter ‘Prime Time’ solutions. This new NCERT book solutions will help students.
Figure it out:
1) At what number is ‘idli-vada’ said for the 10thtime?
Ans: The 10th time ‘idli-vada’ has been said in ‘150’ number.
2) If the game is played for the numbers from 1 till 90, find out:
a) How many times would the children say ‘idli’ (including the times they say ‘idli-vada’)?
Ans: If the game is played from 1 to 90 numbers, then children will say ‘idli’ at the number 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90.
Total 30 times (including the times they say ‘idli-vada’)
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b) How many times would the children say ‘vada’ (including the times they say ‘idli-vada’)?
Ans: The children would say ‘vada’ (including the times they say ‘idli-vada’) is 18 times (5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90).
c) How many times would the children say ‘idli-vada’?
Ans:
The children would say ‘idlivada’ 6 times. (Idli-vada’ is said for multiples of both 3 and 5)
3) What if the game was played till 900? How would your answers change?
Ans:
If the game was played till 900, the answers will also change. Hence, the time to say ‘Idli’, ‘Vada’ and ‘Idli-Vhara’ will also increase.
4.) Is this figure somehow related to the ‘idli-vada’ game?
Ans: Yes, this picture is related to ‘idli-vada’ game.
Hint: Imagine playing the game till 30. Draw the figure if the game is played till 60.
Ans:
Figure it out:
1) Find all multiples of 40 that lie between 310 and 410.
Ans: All multiples of 40 between 310 and 410 are,
310 < 320, 360, 400 > 410
(40 × 8 = 320, 40 × 9 = 360, 40 × 10 = 400)
2) Who am I?
a) I am a number less than 40. One of my factors is 7. The sum of my digits is 8.
Ans:
7 is the common factors of 7, 14, 21, 28, 35, which are less than 40.
And there is one number which have digit sum is 35 = (3+5) = 8.
So, I am 35.
b) I am a number less than 100. Two of my factors are 3 and 5. One of my digits is 1 more than the other.
Ans:
3 and 5 are the common factors of 15, 30, 45, 60, 75, 90, which are less than 100.
And there is one number which one of digit is 1 more than the other that is 45.
So, I am 45.
3) A number for which the sum of all its factors is equal to twice the number is called a perfect number. The number 28 is a perfect number. Its factors are 1, 2, 4,7, 14 and 28. Their sum is 56 which is twice 28. Find a perfect number between 1 and 10.
Ans:
A perfect number between 1 and 10 is 6.
The factors of 6 are: 1, 2, 3 and 6.
Sum of factors is 12 which is twice of 6.
4) Find the common factors of:
| a) 20 and 28
Ans: The factors of 20 are: 1, 2, 4, 5, 10, 20 The factors of 28 are: 1, 2, 4, 7, 14, 28 So, the common factors of 20 and 28 are: 1, 2, and 4. |
b. 35 and 50
Ans: The factors of 35 are: 1, 5, 7, 35 The factors of 50 are: 1, 2, 5, 10, 25, 50 So, the common factors of 35 and 50 are: 1 and 5. |
| c) 4, 8 and 12
Ans: The factors of 4 are: 1, 2, 4 The factors of 8 are: 1, 2, 4, 8 The factors of 12 are: 1, 2, 3, 4, 6, 12 So, the common factors of 4, 6 and 12 are: 1, 2, and 4. |
d. 5, 15 and 25
Ans: The factors of 5 are: 1, 5 The factors of 15 are: 1, 3, 5, 15 The factors of 25 are: 1, 5, 25 So, the common factors of 5, 15 and 25 are: 1 and 5. |
5) Find any three numbers that are multiples of 25 but not multiples of 50.
Ans:
Some numbers that aremultiples of 25 are 25, 50, 75, 100, 125, 150, 175 etc.
Some numbers that are multiples of 50 are 50, 100, 150, 200, 250, 300 etc.
So, the numbers that are multiples of 25 but not multiples of 50 are 25, 75, 125 and 175 etc.
6) Anshu and his friends play the ‘idli-vada’ game with two numbers,which are both smaller than 10. The first time anybody says ‘idlivada’ is after the number 50. What could the two numbers be which are assigned ‘idli’ and ‘vada’?
Ans:
The two numbers which are assigned ‘idli’ and ‘vada’ are 7 and 8 (Both the numbers are smaller than 10).
When it’s multiple of 7 the players should say ‘idli’ and when it’s multiple of 8 the players should say ‘vada’.
(56 is the number which is both the multiples of 7 and 8 when first time idli-vada says after 50).
7) In the treasure hunting game, Grumpy has kept treasures on 28 and 70. What jump sizes will land on both the numbers?
Ans:
The factors of 28 are: 1, 2, 4, 7, 14, and 28. These jump sizes will land on 28
The factors of 70 are: 1, 2, 5, 7, 10, 14, 35, and 70. These jump sizes will land on 70.
1, 2, 7, and 14 are the common factors of 28 and 70.
So, the jump sizes of 1, 2, 7, or 14 will land on both 28 and 70.
8) In the diagram below, Guna has erased all the numbers except the common multiples. Find out what those numbers could be and fill in the missing numbers in the empty regions.
Ans:
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12 and 24
The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48
The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72
1, 2, 3, 4, 6, 8, 12 and 24 are the common factors of 24, 48 and 72
So, the multiples of 6 and 8 are the common multiples of 24, 48, and 72.
9) Find the smallest number that is a multiple of all the numbers from 1 to 10 except for 7.
Ans:
First, we find the L.C.M of all the numbers from 1 to 10 except 7,
We have,
1 = 1, 2 = 1 ×2, 3 = 1 × 3, 4 = 2 × 2, 5 = 1 × 5, 6 = 2 × 3, 8 = 2 × 2 × 2, 9 = 3 × 3, 10 = 2 × 5
So, the L.C.M of these numbers = 1 × 2 × 2 × 2× 3 × 3 × 5 = 360
(Chose the product of highest factors of these numbers)
The smallest number that is a multiple of all the numbers from 1 to 10 except for 7 is 360.
10) Find the smallest number that is a multiple of all the numbers from 1 to 10.
Ans:
First, we find the L.C.M of all the numbers from 1 to 10.
1 = 1, 2 = 1 × 2, 3 = 1 × 3, 4 = 2 × 2, 5 = 1 × 5, 6 = 2 × 3, 7 = 1 × 7, 8 = 2 × 2 × 2, 9 = 3 × 3, 10 = 2 × 5
So, the L.C.M of these numbers = 1 × 2 × 2 × 2 × 3 × 3 × 5 × 7 = 2520
The smallest number that is a multiple of all the numbers from 1 to 10 is 2520.
5.2 Figure it out:
1) We see that 2 is a prime and also an even number. Is there any other even prime?
Ans:
No, 2 is the only prime and even number.
2) Look at the list of primes till 100. What is the smallest difference between two successive primes? What is the largest difference?
Ans:
The smallest difference between two successive prime numbers is 1.
(That is the difference between 2 and 3).
The largest difference between two successive prime numbers is 8.
(That is the difference between 89 and 97).
3) Are there an equal number of primes occurring in every row in the table on the previous page? Which decades have the least number of primes? Which have the most number of primes?
Ans:
No, every row of the table does not have an equal number of primes.
The last row has the least number of primes.
The 1st and 2ndrow has the most number of primes.
4) Which of the following numbers are prime? 23, 51, 37, 26
Ans:
The prime numbers are 23 and 37.
5) Write three pairs of prime numbers less than 20 whose sum is a multiple of 5.
Ans:
The pairs of prime numbers are (2, 3), (3, 7) and (2, 13).
(The pairs of prime numbers are less than 20 and whose sum is a multiple of 5).
6) The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers up to 100.
Ans:
Such pairs of prime numbers are (17 and 71), (37 and 73), and (79 and 97).
7) Find seven consecutive composite numbers between 1 and 100.
Ans:
Seven consecutive composite numbers between 1 and 100 are (90, 91, 92, 93, 94, 95, and 96).
8) Twin primes are pairs of primes having a difference of 2. For example, 3 and 5 are twin primes. So are 17 and 19. Find the other twin primes between 1 and 100.
Ans:
Hence, the other twin primes between 1 and 100 are (5, 7), (11, 13), (41, 43) and (71, 73).
9) Identify whether each statement is true or false. Explain.
a) There is no prime number whose units digit is 4.
Ans:
Yes, there is no prime number whose units digit is 4.
Examples- 134, 564, 374, 974 etc..
Because the units digit place number 4 is divisible by 1, 2, 4.
b) A product of primes can also be prime.
Ans:
A product of primes can’t be prime because the product of primes also divided by the numbers itself.
Example-
A prime number is 7 and the product of primes = 7 × 7 = 49
And 49 is divided by the number 7 itself.
c) Prime numbers do not have any factors.
Ans:
Prime numbers have exactly 2 factors one is 1 and another one is the number itself.
d) All even numbers are composite numbers.
Ans:
No, all even numbers are not composite numbers.
Example – 2 is an even number with 2 factors and also it is a prime number.
Except 2, all even numbers are composite numbers.
e) 2 is a prime and so is the next number, 3. For every otherprime, the next number is composite.
Ans:
Yes it is the only exceptional, for every other prime, the next number is composite.
10) Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330?
Ans:
Find the prime factorization of these numbers,
45 = 3 × 3 × 5
60 = 2 × 2 × 3 × 5
91 = 7 × 13
105 = 3 × 5 × 7
330 = 2 × 3 × 5 × 11
The number is 105 which is the product of three distinct prime numbers.
11) How many three-digit prime numbers can you make using each of 2, 4 and 5 once?
Ans:
When repetition is not allowed 2, 4 and 5 cannot form a single prime number.
Because,
When its units digit is 2 or 4 it is divided by 2, and whenunits digits is 5 it is divided by 5 so that’s why 2, 4 and 5 cannot form a prime number.
12) Observe that 3 is a prime number, and 2 × 3 + 1 = 7 is also a prime. Are there other primes for which doubling and adding 1 gives another prime? Find at least five such examples.
Ans:
Yes, there are other primes whose double and adding 1 gives another prime.
Examples-
i) 2 × 5 + 1 = 11 is a prime.
ii) 2 × 11 + 1 = 23 is a prime.
iii) 2 × 23 + 47 = 47 is a prime.
iv) 2 × 29 + 1 = 59 is a prime.
v) 2 × 41 + 1 = 83 is a prime.
vi) 2 × 53 + 1 = 107 is a prime.
vii) 2 × 89 + 1 = 179 is a prime.
5.4 Figure it out:
1) Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000.
Ans:
The prime factorisations of the following numbers are;
64 = 2 × 2 × 2 × 2 × 2 × 2
104 = 2 × 2 × 2 × 13
105 = 3 × 5 × 7
243 = 3 × 3 × 3 × 3 × 3
320 = 2 × 2 × 2 × 2 × 2 × 2 × 5
141 = 3 × 47
1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
729 = 3 × 3 × 3 × 3 × 3 × 3
1024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
1331 = 11 × 11 × 11
1000 = 2 × 2 × 2 × 5 × 5 × 5
2) The prime factorisation of a number has one 2, two 3s, and one 11. What is the number?
Ans:
Prime factorisation of a number has one 2, two 3s, and one 11
So, the number is = 2 × 3 × 3 × 11 = 198
3) Find three prime numbers, all less than 30, whose product is 1955.
Ans:
Prime factorisation of 1955 = 5 × 17 × 19
So, the three prime numbers are 5, 17 and 19 which all less than 30.
4) Find the prime factorisation of these numbers without multiplying first
a) 56 × 25
Ans:
56 × 25
= 2 × 2 × 2 × 7 × 5 × 5
= 2 × 2 × 2 × 5 × 5 × 7
b) 108 × 75
Ans:
108 × 75
= 2 × 2 × 3 × 3 × 3
c) 1000 × 81
Ans:
1000 × 81
= 2 × 2 × 2 × 5 × 5 × 5 × 3 × 3 × 3 × 3
= 2 × 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5 × 5
5) What is the smallest number whose prime factorisation has:
a) three different prime numbers?
Ans:
The three smallest different prime numbers are 2, 3 and 5.
Product of these numbers = 2 × 3 × 5 = 30
The product of three smallest different prime numbers is always the smallest number whose prime factorisation has three different prime numbers.
So, the number is 30.
b) four different prime numbers?
Ans:
The four smallest different prime numbers are 2, 3, 5 and 7.
Product of these numbers = 2 × 3 × 5 × 7 = 210
The product of four smallest different prime numbers is always the smallest number whose prime factorisation has four different prime numbers.
So, the number is 210.
Figure it out:
1) Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer.
a) 30 and 45
Ans:
The prime factorisation of both numbers are-
30 = 2 × 3 × 5
45 = 3 × 3 × 5
Now, we see that 3 and 5 is a prime factor of 30 as well as 45.
Therefore, 30 and 45 are not co-prime.
b) 57 and 85
Ans:
The prime factorisation of both numbers are-
57 = 3 × 19
85 = 5 × 17
Now, we see that there are no common primes that divide both 57 and 85.
Therefore, 57 and 85 are co-prime.
c) 121 and 1331
Ans:
The prime factorisation of both numbers are-
121 = 11 × 11
1331 = 11 × 11 × 11
Now, we see that 11 is a prime factor of 121 as well as 1331.
Therefore, 121 and 1331 are not co-prime.
d) 343 and 216
Ans:
The prime factorisation of both numbers are-
343 = 7 × 7 × 7
216 = 2 × 2 × 2 × 3 × 3 × 3
Now, we see that there are no common primes that divide both 343 and 216.
Therefore, 343 and 216 are co-prime.
2) Is the first number divisible by the second? Use prime factorisation.
a) 225 and 27
Ans:
The prime factorisations of both numbers-
225 = 3 × 3 × 5 × 5
27 = 3 × 3 × 3
Hence, 5 is a prime factor of 225 but not a prime factor of 27.
Therefore, 225 is not divisible by 27.
b) 96 and 24
Ans:
The prime factorisations of both numbers-
96 = 2 × 2 × 2 × 2 × 2 × 3
24 = 2 × 2 × 2 × 3
Since we can multiply in any order, now it is clear that,
96 = 2 × 2 × 2 × 3 × 2 × 2 = 24 × 8
Therefore, 96 is divisible by 24.
c) 343 and 17
Ans:
The prime factorisations of both numbers-
343 = 7 × 7 × 7
17 = 1 × 17
Hence, 7 is a prime factor of 343 but not a prime factor of 17.
Therefore, 343 is not divisible by 17.
d) 999 and 99
Ans:
The prime factorisations of both numbers-
999 = 3 × 3 × 3 × 37
99 = 3 × 3 × 11
Hence, 11 is a prime factor of 99 but not a prime factor of 999.
Therefore, 999 is not divisible by 99.
3) The first number has prime factorisation 2 × 3 × 7 and the second number has prime factorisation 3 × 7 × 11. Are they co-prime? Does one of them divide the other?
Ans:
Prime factorisation of first number = 2 × 3 × 7
Prime factorisation of second number = 3 × 7 × 11
Now, we see that 3 and 7 is a prime factor of first as well as second number.
Therefore, first and second number is not co-prime.
Hence, 2 is a prime factor of first number but not a prime factor of second number.
Therefore, one of them dose not divided the other number.
4) Guna says, “Any two prime numbers are co-prime”. Is he right?
Ans:
Yes, Guna is right about that.
Because, any two prime numbers are have not any common factors.
Therefore, they are co-prime.
5.5 Figure it out:
1) 2024 is a leap year (as February has 29 days). Leap years occur in the years that are multiples of 4, except for those years that are evenly divisible by 100 but not 400.
a) From the year you were born till now, which years were leapyears?
Ans:
If you were born in 2004, the leap years would be 2004, 2008, 2012, 2016, 2020 and 2024.
To identify the leap years from birth to till now, we need to check which years are divisible by 4 except for those years that are evenly divisible by 100 but not 400.
b) From the year 2024 till 2099, how many leap years are there?
Ans:
The leap years from 2024 to 2099 are;
2024, 2028, 2032, 2036, 2040, 2044, 2048, 2052, 2056, 2060, 2064, 2068, 2072, 2076, 2080, 2084, 2088, 2092 and 2096.
So, there are 19 leap years between 2024 and 2099.
2) Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes.
Ans:
If one reads a number backwards and forwards it remains the same which is called a palindrome.
Largest 4 digit palindrome number divisible by 4;
First we start from the largest palindrome number which is 9999 and then go down and check which is divisible by 4.
So, the Largest 4 digit palindrome divisible by 4 is 8888.
Smallest 4 digit palindrome number divisible by 4;
First we start with the smallest palindrome number which is 1001 and then move upwards and check which is divisible by 4.
So, the smallest 4 digit palindrome divisible by 4 is 2332.
3) Explore and find out if each statement is always true, sometimes true or never true. You can give examples to support your reasoning.
a) Sum of two even numbers gives a multiple of 4.
Ans:
This statement is sometimes true.
Examples-
When it is true,
10 + 14 = 24 is a multiple of 4.
When it is false,
10 + 12 = 22 is not a multiple of 4.
b) Sum of two odd numbers gives a multiple of 4.
Ans:
This statement is sometimes true.
Examples-
When it is true,
7 + 9 = 16 is a multiple of 4.
When it is false,
7 + 11 = 18 is not a multiple of 4.
4) Find the remainders obtained when each of the following numbers are divided by i) 10, ii) 5, iii) 2.
78, 99, 173, 572, 980, 1111, 2345
Ans:
78: (i) 78/10; Remains= 8, (ii) 78/5; Remains= 3, (iii) 78/2; Remains= 0
99: (i) 9, (ii) 4, (iii) 1
173: (i) 3, (ii) 3, (iii) 1
572: (i) 2, (ii) 2, (iii) 0
980: (i) 0, (ii) 0, (iii) 0
1111: (i) 1, (ii) 1, (iii) 1
2345: (i) 5, (ii) 0, (iii) 1
5) The teacher asked if 14560 is divisible by all of 2, 4, 5, 8 and 10. Guna checked for divisibility of 14560 by only two of these numbers and then declared that it was also divisible by all of them. What could those two numbers be?
Ans:
When a number is divisible by 8 it is also divisible by 2 and 4 because 8 is a multiple of both 2 and 4.
And when a number is divisible by 10 it is also divisible by 5 because 10 is a multiple of 5.
So, the two numbers could be 8 and 10. Checking those numbers are sufficient to all the other numbers for 14560.
6) Which of the following numbers are divisible by all of 2, 4, 5, 8 and 10: 572, 2352, 5600, 6000, 77622160.
Ans:
When a number is divisible by 8 and 10 it is also divisible by all the numbers.
Only the last three digits matter when deciding if a given number is divisible by 8 and numbers that are divisible by 10 are those that end with ‘0’.
So, the numbers that’s satisfy all the conditions of being divisible by 2, 4, 5, 8, and 10 are 5600, 6000, and 77622160.
7) Write two numbers whose product is 10000. The two numbers should not have 0 as the units digit.
Ans:
Prime factorisation of 10000 = 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5
So, the two numbers are 16 and 625 whose product is 10000.
(Both the numbers do not have 0 as unit digit).