Concept of LCM and HCF

LCM stands for Lowest Common Multiple, and HCF stands for Highest Common Factor.

The key to telling the difference between these two things is knowing the difference between a multiple and a factor.

A multiple of an integer (whole number) is any integer that appears in its times table. For example, the multiples of 3 are 3, 6, 9, 12, and so on.

A factor of an integer is any integer that divides the integer with no remainder. For example, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

We use LCM and HCF to compare two (or more) integers.

The following diagram shows how to find the HCF and LCM of 24 and 36 using Repeated Division.

preptoz_ctet_math_lcm_hcf

Question / Answer

Question #1 LCM of two prime numbers x and y, (x > y), is 161. The value of 3y – x is

 [Mathematics] [CTET-2014-09]
Options
  • 2
  • -2 correct answer
  • -5
  • 62
Answer Explanation As these are two prime numbers so HCF is 1

And as (x) * (y) = HCF * LCM
so x * y = 1 * 161

x * y = 161

Now as we know these are prime numbers so check the divisors of 161

It is clearly not divisible by 2, 3 & 5

But 7 divides it 161/7 = 23

So two no.s are 7 & 23

As x > y

so x =23 & y = 7

so 3y – x = 3*7 – 23 = 21 – 23 = -2
Question #2 HCF of two numbers is 28 and their LCM is 336. If one  number is 112, then the other number is

[Mathematics] [CTET-2013-07]
Options
  • 56
  • 70
  • 84 correct answer
  • 98
Answer Explanation Using the formula HCF X LCM = number1 x number2
=> 28×336 = 112 x number2
=> number2 = 28×336/112 = 28×3 = 84
Question #3 LCM of 22, 54, 135 and 198 is

[Mathematics] [CTET-2015-02]
Options
  • 22 x 33 x 5 x 11
  • 2 x 33 x 5 x 11 correct answer
  • 22 x 32 x 5 x 11
  • 23 x 32 x 5 x 11
Answer Explanation
2 22 54 135 198
3 11 27 135 99
3 11 9 45 11
3 11 3 15 11
5 11 1 5 11
11 11 1 1 11
1 1 1 1
Question #4 The HCF and LCM of two numbers are 12 and 72 respectively. If the sum of these numbers is 60,then one of the numbers will be

[Mathematics] [UPTET-2017-10]
Options
  • 12
  • 24 correct answer
  • 60
  • 72
Answer Explanation Let two numbers be a & bAs product of two numbers = HCF x LCM

So,    a x b = 12 x 72

And as a + b = 60

So, a x (60-a) = 12 x 72

==> (60a) – (a^2) = 12 x 72
==> (a^2) – (60a) + 12 x 72 = 0

Breaking 60a into 24 & 36, and 72 into 2 & 36
==> (a^2) – (24a + 36a) + 12 x 2 x 36 = 0
==> (a^2) – (24a) – (36a) + 24 x 36 = 0
==> (a^2 – 24a) – (36a – 24 x 36) = 0
==> a(a – 24) – 36(a – 24) = 0
==> (a – 24)(a – 36) = 0

So a can have value either 24 or 36
Question #5 The HCF and LCM of two number are 13 and 1989 respectively. if one of the number is 117 , then other number is

[Mathematics] [UPTET-2017-10]
Options
  • 222
  • 221 correct answer
  • 223
  • 225
Answer Explanation Let the two numbers are a and b

a=117,  b=?
hcf =13
lcm =1989

product of the given numbers = lcm * hcf

a*b =lcm *hcf
117*b= 1989*13
b= 1989*13/117
b=17*13
b=221
therefore; second number =b=221
Question #6 The ratio of two numbers is 5:6 and their HCF is 12. Their LCM is

 [Mathematics] [UPTET-2016-12]
Options
  • 60
  • 72
  • 180
  • 360 correct answer
Answer Explanation LCM = (The part of first no. in ratio) x (The part of second no. in ratio) x HCFSo, LCM = 5 x 6 x 12 = 360
Question #7 What is the LCM of two consecutive odd numbers?

[Mathematics] [OTET-2018-12]
Options
  • the greater odd number
  • the smaller odd number
  • product of two odd numbers correct answer
  • sum of two odd numbers
Question #8 Three numbers are in the ratio 1:2:3. If their HCF is 13, then the LCM of the three numbers is

 [Mathematics] [SCERT-Odisha-BEd-Science-09Aug18-Batch3]
Options
  • 91
  • 78 correct answer
  • 13
  • 39
Answer Explanation LCM = (The part of first no. in ratio) x (The part of second no. in ratio) x (The part of third no. in ratio) x HCF

LCM = 1 x 2 x 3 x 13 = 78
Question #9 Three numbers are in the ratio 1:2:3. If their HCF is 13. then the LCM of the three numbers is

[] [Mathematics] [SCERT-Odisha-BEd-Science-08Aug18-Batch3]
Options
  • 13
  • 39
  • 78 correct answer
  • 91
Answer Explanation LCM = (The part of first no. in ratio) x (The part of second no. in ratio) x (The part of third no. in ratio) x HCF

LCM = 1 x 2 x 3 x 13 = 78
Question #10 If two positive integers a, b expressed as a = pq2 and b = p3q where p, q are prime numbers, then LCM ( a, b) is

 [Mathematics] [SCERT-Odisha-DElEd-Odia-09Aug18-Batch3]
Options
  • p3q3
  • pq
  • p3q2 correct answer
  • p2q2
Answer Explanation As p & q are prime numbersDivisors                     Number 1                     Number 2
p                                   pq2                                   p3q
p                                     q2                                    p2q
p                                     q2                                     pq
q                                     q2                                      q
q                                     q                                         1
1                                        1

Multiply all divisors = p * p * p * q * q = p3q2
Question #11 The LCM of 90 and 144 is

[Mathematics] [SCERT-Odisha-DElEd-Odia-14Aug18-Batch4]
Options
  • 720 correct answer
  • 54
  • 360
  • 18
Answer Explanation Divisor                    Number 1                    Number  2
2                                   90                                    144
2                                   45                                    72
2                                   45                                    36
2                                   45                                    18
3                                   45                                    9
3                                   15                                    3
5                                   5                                       1
1                                       1

So, LCM = 2 x 2 x 2 x 2 x 3 x 3 x 5 = 720
Question #12 The product of LCM and HCF of two numbers is 7605. If one number is 117 then other number is

 [Mathematics] [SCERT-Odisha-DElEd-Odia-13Aug18-Batch1]
Options
  • 85
  • 13
  • 17
  • 65 correct answer
Answer Explanation As Number 1 x Number 2 = HCF x LCM

So, 117 x Number 2 = 7605

Number 2 = 7605/117 = 65
Question #13 If a = 8×3, b = 2×3×5, c = 3n×5 and LCM (abc) is 8×32×5 then what is the value of ‘n’?

 [Mathematics] [SCERT-Odisha-DElEd-Odia-13Aug18-Batch2]
Options
  • 4
  • 5
  • 2 correct answer
  • 3
Answer Explanation LCM would be formed by taking highest power of each factor from each no.

As,
a = 8 x 3 = 23 x 3
b = 2 x 3 x 5
c = 3n×5

Highest power of 2 = 3
Highest power of 3 = n
Highest power of 5 = 1

So, LCM = 23 x 3n×5

Comparing to LCM = 8 x 32 x 5 = 23 x 32 x 5

So,
n = 2
Question #14 Two numbers are in the ratio 3:4. The product of their HCF and LCM is 2028. The sum of the numbers will be

[Mathematics] [UPTET-2018-11]
Options
  • 86
  • 68
  • 72
  • 91 correct answer
Answer Explanation LCM = (The part of first no. in ratio) x (The part of second no. in ratio) x HCF = 3 x 4 x HCF

Product of two no.s = HCF x LCM = 2028

So,
HCF x [3 x 4 x HCF] = 2028

HCF2  = 2028/12 = 169

HCF = 13

Number 1 = 3 x HCF = 3 x 13 = 39
Number 2 = 4 x HCF = 4 x 13 = 52

Sum of Numbers = 39 + 52 = 91
Question #15 The HCF of 14, 21 and another number is 7 and their LCM is 210. Then what is the third number?

 [Mathematics] [OTET-2017-09]
Options
  • 25
  • 28
  • 35 correct answer
  • 105
Answer Explanation Let third number be yIf we divide all numbers and their LCM by HCF then  ==> 14/7, 21/7, y/7 & 210/7

Now,

(14/7) x (21/7) x (y/7) = (210/7)

==> 2 x 3 x (y/7) = 30

==> (y/7) = 5

==> y = 35
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