What is the HCF of (16) and (27)?
Correct answer: A
Step 1: (16=2^4) and (27=3^3). Step 2: They have no common prime factor. Step 3: So the HCF is (1), and these numbers are co-prime.
Mathematics
HCF and LCM using prime factorisation
अभाज्य गुणनखंडन द्वारा महत्तम समापवर्तक और लघुत्तम समापवर्त्य
In this Class 10 Mathematics topic from Real Numbers, students learn to find the HCF and LCM of two or more numbers through prime factorisation. They break each number into prime factors, compare the resulting powers, and select the appropriate common factors: the lowest powers for HCF and the highest powers for LCM. The topic also develops accuracy in writing factor trees, simplifies comparison between methods, and helps students apply these concepts to numerical and everyday problem-solving situations.
TOPIC PRACTICE
Quiz this set
Up to 25 questions from this page. Select your focus, then start.
25 questions
Choose questions
Easy · Level 4
View options
- (1)
- (2)
- (3)
- (9)
Easy · Level 4
View options
- (144)
- (216)
- (324)
- (432)
Easy · Level 4
View options
- (29)
- (90)
- (180)
- (360)
Easy · Level 4
View options
- (3)
- (13)
- (26)
- (39)
Easy · Level 4
View options
- (78)
- (117)
- (156)
- (234)
Easy · Level 4
View options
- (8)
- (12)
- (16)
- (24)
Easy · Level 4
View options
- (7)
- (14)
- (21)
- (28)
Easy · Level 4
View options
- (30)
- (45)
- (60)
- (90)
Easy · Level 4
View options
- (24)
- (32)
- (48)
- (96)
Easy · Level 4
View options
- (6)
- (12)
- (18)
- (24)
Easy · Level 4
View options
- (9)
- (18)
- (27)
- (36)
Easy · Level 4
View options
- (5)
- (10)
- (25)
- (50)
Easy · Level 4
View options
- (150)
- (200)
- (300)
- (400)
Easy · Level 4
View options
- (7)
- (14)
- (21)
- (49)
Easy · Level 4
View options
- (294)
- (441)
- (588)
- (882)
Easy · Level 4
View options
- (18)
- (36)
- (54)
- (72)
Easy · Level 4
View options
- (270)
- (360)
- (540)
- (1080)
Easy · Level 4
View options
- (9)
- (18)
- (27)
- (54)
Easy · Level 4
View options
- (81)
- (108)
- (162)
- (324)
Easy · Level 4
View options
- (12)
- (24)
- (36)
- (48)
Easy · Level 4
View options
- (240)
- (360)
- (720)
- (1440)
Easy · Level 4
View options
- (180)
- (240)
- (360)
- (720)
Easy · Level 4
View options
- (600)
- (900)
- (1200)
- (1800)
Easy · Level 4
View options
- Smaller power of common factors
- Highest power of all prime factors
- Only common factors
- Only the smallest number
Easy · Level 4
View options
- ( \text{HCF}+\text{LCM}=\text{product} )
- ( \text{HCF}\times\text{LCM}=\text{product of the two numbers} )
- ( \text{HCF}=\text{LCM} )
- ( \text{LCM}=1 )
What is the LCM of (16) and (27)?
Correct answer: D
Step 1: (16=2^4) and (27=3^3). Step 2: There is no common factor, so the LCM is their product. Step 3: (16\times27=432), so the answer is (432).
The two co-prime numbers are (9) and (20). What will be their LCM?
Correct answer: C
Step 1: (9=3^2) and (20=2^2\times5). Step 2: They have no common prime factor. Step 3: The LCM of co-prime numbers is their product, so (9\times20=180).
What will be the HCF of (26) and (39)?
Correct answer: B
Step 1: (26=2\times13) and (39=3\times13). Step 2: The common prime factor is (13). Step 3: Therefore, the HCF is (13).
What will be the LCM of (26) and (39)?
Correct answer: A
Step 1: (26=2\times13) and (39=3\times13). Step 2: For LCM, take (2), (3), and (13). Step 3: (2\times3\times13=78), so the answer is (78).
A class has (32) boys and (48) girls. What can be the maximum number of equal groups?
Correct answer: C
Step 1: For maximum equal groups, find the HCF. Step 2: (32=2^5) and (48=2^4\times3), so HCF (=2^4=16). Step 3: In equal distribution questions, the word maximum usually points to HCF.
A shopkeeper wants to pack (42) apples and (56) oranges in equal bags. What is the maximum number of bags possible?
Correct answer: B
Step 1: For the maximum number of equal bags, find the HCF of (42) and (56). Step 2: (42=2\times3\times7) and (56=2^3\times7), so HCF (=2\times7=14). Step 3: In packing questions with equal and maximum, use HCF.
Two bells ring at intervals of (15) minutes and (20) minutes. After how many minutes will they ring together again?
Correct answer: C
Step 1: The next common ringing time is the LCM of the intervals. Step 2: (15=3\times5) and (20=2^2\times5), so LCM (=2^2\times3\times5=60). Step 3: For repeated time events, use LCM.
Three indicators glow at intervals of (8), (12), and (16) seconds. After how many seconds will they glow together again?
Correct answer: C
Step 1: The next common glowing time is the LCM of (8), (12), and (16). Step 2: (8=2^3), (12=2^2\times3), and (16=2^4). Step 3: (2^4\times3=48), so the answer is (48) seconds.
Ropes of (60) m and (84) m are to be cut into equal pieces of maximum length. What will be the length of each piece?
Correct answer: B
Step 1: For equal pieces of maximum length, find the HCF. Step 2: (60=2^2\times3\times5) and (84=2^2\times3\times7), so HCF (=2^2\times3=12). Step 3: In cutting questions, maximum equal length is found by HCF.
Strips of (54) m and (90) m are to be cut into equal pieces of maximum length. What will be the length of each piece?
Correct answer: B
Step 1: For maximum equal length, find the HCF of (54) and (90). Step 2: (54=2\times3^3) and (90=2\times3^2\times5), so HCF (=2\times3^2=18). Step 3: When lengths are cut equally, use HCF.
What is the HCF of (75) and (100)?
Correct answer: C
Step 1: (75=3\times5^2) and (100=2^2\times5^2). Step 2: The common prime factor is (5), and the smaller power is (5^2). Step 3: (5^2=25), so the HCF is (25).
What is the LCM of (75) and (100)?
Correct answer: C
Step 1: (75=3\times5^2) and (100=2^2\times5^2). Step 2: The highest powers are (2^2), (3), and (5^2). Step 3: (4\times3\times25=300), so the LCM is (300).
What is the HCF of (98) and (147)?
Correct answer: D
Step 1: (98=2\times7^2) and (147=3\times7^2). Step 2: The common prime factor is (7), and the smaller power is (7^2). Step 3: (7^2=49), so the HCF is (49).
What is the LCM of (98) and (147)?
Correct answer: A
Step 1: (98=2\times7^2) and (147=3\times7^2). Step 2: The LCM includes (2), (3), and (7^2). Step 3: (2\times3\times49=294), so the answer is (294).
If (108=2^2\times3^3) and (180=2^2\times3^2\times5), what is their HCF?
Correct answer: B
Step 1: The common prime factors are (2) and (3). Step 2: The smaller powers are (2^2) and (3^2). Step 3: (4\times9=36), so the HCF is (36).
If (108=2^2\times3^3) and (180=2^2\times3^2\times5), what is their LCM?
Correct answer: C
Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^2), (3^3), and (5). Step 3: (4\times27\times5=540), so the answer is (540).
What is the HCF of (27), (54), and (81)?
Correct answer: C
Step 1: (27=3^3), (54=2\times3^3), and (81=3^4). Step 2: The common prime factor is (3), and the smallest power is (3^3). Step 3: (3^3=27), so the HCF is (27).
What is the LCM of (27), (54), and (81)?
Correct answer: C
Step 1: (27=3^3), (54=2\times3^3), and (81=3^4). Step 2: The highest powers are (2) and (3^4). Step 3: (2\times81=162), so the LCM is (162).
What is the HCF of (72) and (120)?
Correct answer: B
Step 1: (72=2^3\times3^2) and (120=2^3\times3\times5). Step 2: The smaller powers of common factors are (2^3) and (3). Step 3: (8\times3=24), so the HCF is (24).
What is the LCM of (72) and (120)?
Correct answer: B
Step 1: (72=2^3\times3^2) and (120=2^3\times3\times5). Step 2: The highest powers are (2^3), (3^2), and (5). Step 3: (8\times9\times5=360), so the LCM is (360).
The prime factorisation of a number is (2^3\times3^2\times5). What is the number?
Correct answer: C
Step 1: First evaluate the powers: (2^3=8) and (3^2=9). Step 2: Now (8\times9\times5=360). Step 3: While forming a number from factorisation, simplify powers first and then multiply.
The prime factorisation of a number is (2^4\times3\times5^2). What is the number?
Correct answer: C
Step 1: (2^4=16) and (5^2=25). Step 2: The number is (16\times3\times25=1200). Step 3: In long multiplication, pair easy numbers first, such as (16\times25=400).
While finding LCM by prime factorisation, which power is taken?
Correct answer: B
Step 1: LCM is the smallest number divisible by all the given numbers. Step 2: Therefore, the highest powers of all prime factors are taken. Step 3: Remember the rule: all factors with highest powers for LCM.
Which statement about HCF and LCM is correct for two numbers?
Correct answer: B
Step 1: For two numbers, a special relation is used. Step 2: The product of HCF and LCM equals the product of the two numbers. Step 3: Apply this relation directly only for two numbers.
QUIZ COMPLETE