Maharashtra Board Class 7 Math Solution Chapter 3 – HCF and LCM
Balbharati Maharashtra Board Class 7 Math Solution Chapter 3: HCF and LCM. Marathi or English Medium Students of Class 7 get here HCF and LCM full Exercise Solution.
Std |
Maharashtra Class 7 |
Subject |
Mathematics Solution |
Chapter |
HCF and LCM |
Practice Set 10
1.) Which number is neither a prime number nor a composite number?
ANS:
1 Is the number which is neither a prime number nor a composite number.
2.) Which of the following are pairs of co-primes?
(i) 8, 14
ANS:
Co-prime number is the numbers having only 1 as a common factor.
8, 14 have 1and 2 common factors.
Hence this is not a pair of Co-prime number.
(ii) 4, 5
ANS:
Co-prime number is the numbers having only 1 as a common factor.
4, 5 have only 1 common factor.
Hence this is a pair of Co-prime number.
(iii) 17, 19
ANS:
Co-prime number is the numbers having only 1 as a common factor.
17, 19 is a prime numbers having only 1 common factor.
Hence this is a pair of Co-prime number.
(iv) 27, 15
ANS:
Co-prime number is the numbers having only 1 as a common factor.
27, 15 have 1and 3 common factors
Hence this is not a pair of Co-prime number.
3.) List the prime numbers from 25 to 100 and say how many they are.
ANS:
Prime number is a number which cannot divide by any number except 1.
Prime numbers from 25 to 100 = 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
There are total 16 prime numbers from 25 to 100.
4.) Write all the twin prime numbers from 51 to 100.
ANS:
Twin prime number is a prime numbers having difference between them is 2.
Twin prime numbers from 51 to 100 = There are 2 pairs of twin prime numbers from 51 to 100.
59 and 61, 71 and 73 are the pairs of twin prime numbers.
5.) Write 5 pairs of twin prime numbers from 1 to 50.
ANS:
There are total 8 pairs of twin prime numbers from 1 to 100.
Twin prime number is a prime numbers having difference between them is 2.
5 pairs of twin prime numbers from 1 to 50 = (3,5) ,(5,7) ,(11,13),(17,19),(29,31),(41,43)
6.) Which are the even prime numbers?
ANS:
2 is the only number which is even prime number.
Practice Set 11
¤ Factorise the following numbers into primes.
(i) 32
ANS:
We have to divide given number as a product of prime numbers.
We splitting 32 as a multiplication of 2 × 2 × 2 × 2 × 2
32 = 2 × 2 × 2 × 2 × 2
(ii) 57
ANS:
We have to divide given number as a product of prime numbers.
We splitting 57 as a multiplication of 3 × 19
57 = 3 × 19
(iii) 23
ANS:
We have to divide given number as a product of prime numbers.
(iv) 150
ANS:
We have to divide given number as a product of prime numbers.
We splitting 150 as a multiplication of 2 × 3 × 5 × 5
150 = 2 × 3 × 5 × 5
(v) 216
ANS:
We have to divide given number as a product of prime numbers.
‘
We splitting 216 as a multiplication of 2 × 2 × 2 × 3 × 3 × 3
216 = 2 × 2 × 2 × 3 × 3 × 3
(vi) 208
ANS:
We have to divide given number as a product of prime numbers.
We splitting 208 as a multiplication of 2 × 2 × 2 × 2 × 13
208 = 2 × 2 × 2 × 2 × 13
(vii) 765
ANS:
We have to divide given number as a product of prime numbers.
We splitting 765 as a multiplication of 3 × 3 × 5 × 17
765 = 3 × 3 × 5 × 17
(viii) 342
ANS:
We have to divide given number as a product of prime numbers.
We splitting 342 as a multiplication of 2 × 3 × 3 × 19
342 = 2 × 3 × 3 × 19
(ix) 377
ANS:
We have to divide given number as a product of prime numbers.
We splitting 377 as a multiplication of 13 × 29
377 = 13 × 29
(x) 559
ANS:
We have to divide given number as a product of prime numbers.
We splitting 559 as a multiplication of 13 × 43
559 = 13 × 43
Practice Set 12
1.) Find the HCF.
(i) 25, 40
ANS:
HCF is a Highest Common Factor.
In HCF we factorise both numbers and taking common factor between them and multiply it.
Multiplication of common factor of both number is called as Highest Common Factor.
Factor of 25 = 5 x 5
Factor of 40 = 2 x 2 x 2 x 5
The common factor between them is 5 hence
Highest Common Factor of 25, 40 is 5.
(ii) 56, 32
ANS:
HCF is a Highest Common Factor.
In HCF we factorise both numbers and taking common factor between them and multiply it.
Multiplication of common factor of both number is called as Highest Common Factor.
Factor of 56 = 2 x 2 x 2 x 7
Factor of 32 = 2 x 2 x 2 x 2 x 2
The common factor between them is 2 x 2 x 2
Highest Common Factor of 56, 32 is 8.
(iii) 40, 60, 75
ANS:
HCF is a Highest Common Factor.
In HCF we factorise both numbers and taking common factor between them and multiply it.
Multiplication of common factor of both number is called as Highest Common Factor.
Factor of 40 = 2 x 2 x 2 x 5
Factor of 60 = 2 x 2 x 3 x 5
Factor of 75 = 3 x 5 x 5
The common factor between them is 5
Highest Common Factor of 40, 60, 75 is 5.
(iv) 16, 27
ANS:
HCF is a Highest Common Factor.
In HCF we factorise both numbers and taking common factor between them and multiply it.
Multiplication of common factor of both number is called as Highest Common Factor.
Factor of 16 = 2 x 2 x 2 x 2
Factor of 27 = 3 X 3 X 3
There is no common factor between them
Highest Common Factor of 16, 27 is 1.
(v) 18, 32, 48
ANS:
HCF is a Highest Common Factor.
In HCF we factorise both numbers and taking common factor between them and multiply it.
Multiplication of common factor of both number is called as Highest Common Factor.
Factor of 18 = 2 x 3 x 3
Factor of 32 = 2 x 2 x 2 x 2 x 2
Factor of 48 = 2 x 2 x 2 x 2 x 3
The common factor between them is 2.
Highest Common Factor of 18, 32, 48 is 2.
(vi) 105, 154
ANS:
HCF is a Highest Common Factor.
In HCF we factorise both numbers and taking common factor between them and multiply it.
Multiplication of common factor of both number is called as Highest Common Factor.
Factor of 105 = 3 x 5 x 7
Factor of 154 = 2 x 7 x 11
The common factor between them is 7.
Highest Common Factor of 105, 154 is 7.
(vii) 42, 45, 48
ANS:
HCF is a Highest Common Factor.
In HCF we factorise both numbers and taking common factor between them and multiply it.
Multiplication of common factor of both number is called as Highest Common Factor.
Factor of 42 = 2 x 3 x 7
Factor of 45 = 3 x 3 x 5
Factor of 48 = 2 x 2 x 2 x 2 x 3
The common factor between them is 3.
Highest Common Factor of 42, 45, 48 is 3.
(viii) 57, 75, 102
ANS:
HCF is a Highest Common Factor.
In HCF we factorise both numbers and taking common factor between them and multiply it.
Multiplication of common factor of both number is called as Highest Common Factor.
Factor of 57 = 3 x 19
Factor of 75 = 3 x 25
Factor of 102 = 2 x 3 x 17
The common factor between them is 3.
Highest Common Factor of 57, 75, 102 is 3.
(ix) 56, 57
ANS:
HCF is a Highest Common Factor.
In HCF we factorise both numbers and taking common factor between them and multiply it.
Multiplication of common factor of both number is called as Highest Common Factor.
Factor of 56 = 2 x 2 x 2 x 7
Factor of 57 = 3 x 19
There is no common factor between them
Highest Common Factor of 56,57 is 1.
(x) 777, 315, 588
ANS:
HCF is a Highest Common Factor.
In HCF we factorise both numbers and taking common factor between them and multiply it.
Multiplication of common factor of both number is called as Highest Common Factor.
Factor of 777 = 3 x 259
Factor of 315 = 3 x 105
Factor of 588 = 3 x 196
The common factor between them is 3.
Highest Common Factor of 777, 315, 588 is 3.
2.) Find the HCF by the division method and reduce to the simplest form.
(i) 275/ 525
ANS:
HCF is a Highest Common Factor.
In HCF we factorise both numbers and taking common factor between them and multiply it.
Multiplication of common factor of both number is called as Highest Common Factor.
We are factorise this fraction by common factor.
275/ 525 = 25 x 11 / 25 x 21
Common factor of fraction is 25.
Hence HCF is 25.
Simplest form of fraction = (275/25) / (525 / 25) = 11/21
(ii) 76 /133
ANS:
HCF is a Highest Common Factor.
In HCF we factorise both numbers and taking common factor between them and multiply it.
Multiplication of common factor of both number is called as Highest Common Factor.
We are factorise this fraction by common factor.
76 /133 = 19 x 4 / 19 x 7
Common factor of fraction is 19.
Hence HCF is 19.
Simplest form of fraction = (76/19) /(133/19) = 4/7
(iii) 161/ 6
ANS:
HCF is a Highest Common Factor.
In HCF we factorise both numbers and taking common factor between them and multiply it.
Multiplication of common factor of both number is called as Highest Common Factor.
We are factorise this fraction by common factor.
161/ 6 = 161 x 1 / 6 x 1
There is no common factor between them
Hence HCF is 1.
Simplest form of fraction = 161/ 6
Practice Set 13
1.) Find the LCM.
(i) 12, 15
ANS:
LCM is Least Common Multiple.
The Least Common Multiple of the given numbers is the smallest number that is divisible by each of the given numbers.
I.e. Multiplication of common and uncommon factors gives Least Common Multiple.
Multiplication of common and uncommon factors gives Least Common Multiple.
LCM of 12, 15 = 3 x 4 x 5 = 60
LCM of 12, 15 = 60
(ii) 6, 8, 10
ANS:
LCM is Least Common Multiple.
The Least Common Multiple of the given numbers is the smallest number that is divisible by each of the given numbers.
I.e. Multiplication of common and uncommon factors gives Least Common Multiple.
Multiplication of common and uncommon factors gives Least Common Multiple.
LCM of 6, 8, 10 = 2 x 3 x 4 x 5 = 120
LCM of 6, 8, 10 = 120
(iii) 18, 32
ANS:
LCM is Least Common Multiple.
The Least Common Multiple of the given numbers is the smallest number that is divisible by each of the given numbers.
I.e. Multiplication of common and uncommon factors gives Least Common Multiple.
Multiplication of common and uncommon factors gives Least Common Multiple.
LCM of 18, 32 = 2 x 9 x 16 = 288
LCM of 18, 32 = 288
(iv) 10, 15, 20
ANS:
LCM is Least Common Multiple.
The Least Common Multiple of the given numbers is the smallest number that is divisible by each of the given numbers.
I.e. Multiplication of common and uncommon factors gives Least Common Multiple.
Multiplication of common and uncommon factors gives Least Common Multiple.
LCM of 10, 15, 20 = 5 x 2 x 2 x 3
LCM of 10, 15, 20 = 60
(v) 45, 86
ANS:
LCM is Least Common Multiple.
The Least Common Multiple of the given numbers is the smallest number that is divisible by each of the given numbers.
I.e. Multiplication of common and uncommon factors gives Least Common Multiple.
Here in 45, 86 there is no common factor between them.
Hence LCM is multiplication of that numbers.
LCM = 45 x 86
LCM of 45, 86 = 3870
(vi) 15, 30, 90
ANS:
LCM is Least Common Multiple.
The Least Common Multiple of the given numbers is the smallest number that is divisible by each of the given numbers.
I.e. Multiplication of common and uncommon factors gives Least Common Multiple.
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 15, 30, 90 = 15 x 2 x 3
LCM of 15, 30, 90 = 90
(vii) 105, 195
ANS:
LCM is Least Common Multiple.
The Least Common Multiple of the given numbers is the smallest number that is divisible by each of the given numbers.
I.e. Multiplication of common and uncommon factors gives Least Common Multiple.
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 105, 195 = 15 x 7 x 13
LCM of 105, 195 = 1365
(viii) 12, 15, 45
ANS:
LCM is Least Common Multiple.
The Least Common Multiple of the given numbers is the smallest number that is divisible by each of the given numbers.
I.e. Multiplication of common and uncommon factors gives Least Common Multiple.
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 12, 15, 45 = 3 x 5 x 4 x 3
LCM of 12, 15, 45 = 180
(ix) 63, 81
ANS:
LCM is Least Common Multiple.
The Least Common Multiple of the given numbers is the smallest number that is divisible by each of the given numbers.
I.e. Multiplication of common and uncommon factors gives Least Common Multiple.
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 63, 81 = 7x 9 x 9
LCM of 63, 81 = 567
(x) 18, 36, 27
ANS:
LCM is Least Common Multiple.
The Least Common Multiple of the given numbers is the smallest number that is divisible by each of the given numbers.
I.e. Multiplication of common and uncommon factors gives Least Common Multiple.
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 18, 36, 27 = 9 x 2 x 2 x 3
LCM of 18, 36, 27 = 108
2.) Find the HCF and LCM of the numbers given below. Verify that their product is equal to the product of the given numbers.
(i) 32, 37
ANS:
We have to find HCF and LCM of the numbers 32, 37.
Here in 32, 37 there is no common factor between them.
When there is no common factor between them Hcf is always 1.
And when there is no common factor LCM is product of that numbers.
HCF of 32, 37 = 1
LCM of 32, 37 = 32 x 37
LCM of 32, 37 = 1184
(ii) 46, 51
ANS:
We have to find HCF and LCM of the numbers 46, 51.
Here in 46, 51 there is no common factor between them.
When there is no common factor between them Hcf is always 1.
And when there is no common factor LCM is product of that numbers.
HCF of 46, 51 = 1
LCM of 46, 51 = 46 x 51
LCM of 46, 51 = 2346
(iii) 15, 60
ANS:
HCF is the multiplication of common factor.
Here 15 is the common factor.
HCF of 15, 60 = 15
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 15, 60 = 15 x 4
LCM of 15, 60 = 60
(iv) 18, 63
ANS:
HCF is the multiplication of common factor.
Here 9 is the common factor.
HCF of 18, 63 = 9
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 18, 63 = 9 x 7
LCM of 18, 63 = 63
(v) 78, 104
ANS:
HCF is the multiplication of common factor.
Here 13 and 2 is the common factor.
HCF of 78,104 = 13 x 2 = 26
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 78,104 = 13 x 2 x 3 x 4
LCM of 78,104 = 312
Practice Set 14
1.) Choose the right option.
(i) The HCF of 120 and 150 is………………..
(1) 30 (2) 45 (3) 20 (4) 120
ANS:
HCF is the multiplication of common factor.
Here 30 is the common factor.
The HCF of 120 and 150 is 30.
(1) 30
(ii) The HCF of this pair of numbers is not 1.
(1) 13, 17 (2) 29, 20 (3) 40, 20 (4) 14, 15
ANS:
The HCF is 1 only when there is no common factor between them.
Here, 40, 20 is divisible by 2, 4 ,10 and 20.
Hence HCF of 40, 20 is not 1.
(3) 40, 20
2.) Find the HCF and LCM.
(i) 14, 28
ANS:
HCF is the multiplication of common factor.
Here 14 is the common factor.
HCF of 14, 28= 14
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 14, 28 = 14 x 2
LCM of 14, 28= 28
(ii) 32, 16
ANS:
HCF is the multiplication of common factor.
Here 16 is the common factor.
HCF of 32, 16= 16
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of32, 16 = 16 x 2
LCM of 32, 16= 32
(iii) 17, 102, 170
ANS:
HCF is the multiplication of common factor.
Here 17 is the common factor.
HCF of 17, 102, 170 = 17
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 17, 102, 170 = 17 x 2 x 3 x 5
LCM of 17, 102, 170 = 510
(iv) 23, 69
ANS:
HCF is the multiplication of common factor.
Here 23 is the common factor.
HCF of 23, 69 = 23
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 23, 69 = 23 x3
LCM of 23, 69 = 69
(v) 21, 49, 84
ANS:
HCF is the multiplication of common factor.
Here 7 is the common factor.
HCF of 21, 49, 84 = 7
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 21, 49, 84= 7 x 3 x 7 x 4
LCM of 21, 49, 84 = 588
3.) Find the LCM.
(i) 36, 42
ANS:
LCM is Least Common Multiple.
The Least Common Multiple of the given numbers is the smallest number that is divisible by each of the given numbers.
I.e. Multiplication of common and uncommon factors gives Least Common Multiple.
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 36, 42 = 6 x 6 x 7
LCM of 36, 42 = 252
(ii) 15, 25, 30
ANS:
LCM is Least Common Multiple.
The Least Common Multiple of the given numbers is the smallest number that is divisible by each of the given numbers.
I.e. Multiplication of common and uncommon factors gives Least Common Multiple.
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 15, 25, 30 = 5 x 3 x 5 x 2
LCM of 15, 25, 30 = 150
(iii) 18, 42, 48
ANS:
LCM is Least Common Multiple.
The Least Common Multiple of the given numbers is the smallest number that is divisible by each of the given numbers.
I.e. Multiplication of common and uncommon factors gives Least Common Multiple.
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 18, 42, 48 = 6 x 3 x 7 x 8
LCM of 18, 42, 48 = 1008
(iv) 4, 12, 20
ANS:
LCM is Least Common Multiple.
The Least Common Multiple of the given numbers is the smallest number that is divisible by each of the given numbers.
I.e. Multiplication of common and uncommon factors gives Least Common Multiple.
Multiplication of common and uncommon factors gives Least Common Multiple
LCM of 4, 12, 20 = 4 x 3 x 5
LCM of 4, 12, 20 = 60
4.) The LCM and HCF of two numbers are 432 and 72 respectively. If one of the numbers is 216, what is the other?
ANS:
Here LCM and HCF of two numbers are given. 432 and 72
We know,
One number x another number = LCM x HCF
Here, one of the numbers is 216
216 x another number = 432 x 72
Another number =432 x 72 /216
Another number = 432 / 3
Another number = 144
The other number is 144.
5) The product of two two-digit numbers is 765 and their HCF is 3. What is their LCM?
ANS:
Given that the product of two two-digit numbers is 765 and their HCF is 3.
We know,
One number x another number = LCM x HCF
765 = LCM x HCF
765 =LCM x 3
LCM = 765/3
LCM = 255
LCM is 255.
6) Which two consecutive even numbers have an LCM of 180?
ANS:
When there is two consecutive even numbers there HCF is always 2.
Let
X be the first even number then next even number is (x + 2)
We know,
One number x another number = LCM x HCF
X x(x + 2) = LCM x HCF
Here LCM is 180
HCF is 2
X x (x + 2) = 180 x 2
X^{2 }+ 2x = 360
X^{2 }+ 2x -360 = 0
By solving this equation we get,
X = 18
Then next consecutive even number is ( x + 2 ) = 20.