What is the correct prime factorisation of 720?
Correct answer: A
Step 1: Write (720=72\times10). Step 2: (72=2^3\times3^2) and (10=2\times5), so (720=2^4\times3^2\times5). Step 3: Do not keep composite factors in the final answer.
Mathematics
Fundamental Theorem of Arithmetic
अंकगणित का मौलिक प्रमेय
In this Class 10 Mathematics topic from the chapter Real Numbers, students learn that every integer greater than 1 can be expressed as a product of prime numbers, and that this prime factorisation is unique apart from the order of the factors. They practise finding prime factors and use the theorem to understand and determine the HCF and LCM of numbers. The topic builds clear reasoning about the structure of whole numbers and supports later work with divisibility and number relationships.
TOPIC PRACTICE
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Medium · Level 3
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- (2^4\times3^2\times5)
- (2^3\times3^3\times5)
- (2^4\times3\times5^2)
- (72\times10)
Medium · Level 3
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- (2^2\times3\times7\times11)
- (2\times3^2\times7\times11)
- (2^2\times3\times7^2)
- (84\times11)
Medium · Level 3
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- (2^3\times3\times7^2)
- (2^2\times3^2\times7^2)
- (2^3\times3^2\times7)
- (24\times49)
Medium · Level 3
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- 120
- 240
- 360
- 840
Medium · Level 3
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- 25200
- 50400
- 75600
- 100800
Medium · Level 3
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- 90
- 100
- 120
- 140
Medium · Level 3
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- 8640
- 384
- 336
- 15
Medium · Level 3
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- 504
- 756
- 1008
- 1512
Medium · Level 3
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- 2
- 3
- 4
- 5
Medium · Level 3
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- (3^2\times5^2\times7)
- (3^3\times5\times7)
- (3\times5^3\times7)
- (15\times105)
Medium · Level 3
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- 360
- 540
- 720
- 1440
Medium · Level 3
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- 2
- 3
- 4
- 6
Medium · Level 3
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- They are co-prime
- Their HCF is 3
- Their common prime factor is 2
- Their LCM is 64
Medium · Level 3
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- 17
- 23
- 391
- 1
Medium · Level 3
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- (2^3\times3\times13)
- (8\times3\times13)
- (24\times13)
- (2\times12\times13)
Medium · Level 3
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- 12
- 24
- 48
- 72
Medium · Level 3
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- 504
- 1008
- 1512
- 2016
Medium · Level 3
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- (2\times3^3\times5\times7)
- (2^2\times3^2\times5\times7)
- (2\times3^2\times5\times7)
- (189\times10)
Medium · Level 3
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- 288
- 576
- 1152
- 1728
Medium · Level 3
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- 5
- 7
- 9
- 11
Medium · Level 3
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- 1
- 2
- 3
- 4
Medium · Level 3
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- 1
- 2
- 3
- 4
Medium · Level 3
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- They are co-prime
- Their HCF is 7
- Their common prime factor is 3
- Their LCM is 1
Medium · Level 3
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- 73
- 315
- 630
- 1260
Medium · Level 3
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- (2^{11})
- (2^{10})
- (4^6)
- (32^3)
Which is the prime factorisation of 924?
Correct answer: A
Step 1: Write (924=84\times11). Step 2: (84=2^2\times3\times7), so (924=2^2\times3\times7\times11). Step 3: 84 is composite, so do not leave it in the final prime form.
What is the correct prime factorisation of 1176?
Correct answer: A
Step 1: Write (1176=24\times49). Step 2: (24=2^3\times3) and (49=7^2), so (1176=2^3\times3\times7^2). Step 3: 24 and 49 are composite, so write prime powers in the final form.
If (a=2^5\times3^2\times5) and (b=2^3\times3\times5^2\times7), what is the HCF of (a) and (b)?
Correct answer: A
Step 1: For HCF, take the smaller powers of common prime factors. Step 2: The smaller powers are (2^3), (3^1), and (5^1). Step 3: (8\times3\times5=120), so the answer is 120.
If (a=2^5\times3^2\times5) and (b=2^3\times3\times5^2\times7), what is the LCM of (a) and (b)?
Correct answer: B
Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^5), (3^2), (5^2), and (7). Step 3: (32\times9\times25\times7=50400), so the answer is 50400.
If the product of two numbers is 2520 and their HCF is 21, what is their LCM?
Correct answer: C
Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: LCM (=2520\div21=120). Step 3: Use this formula directly only for two numbers.
If the HCF of two numbers is 24 and their LCM is 360, what is their product?
Correct answer: A
Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: (24\times360=8640). Step 3: Apply the HCF-LCM relation carefully.
If (x=2^4\times3^2\times7), what is the value of (x)?
Correct answer: C
Step 1: First calculate (2^4=16) and (3^2=9). Step 2: (16\times9\times7=1008). Step 3: Simplify powers first while finding the number.
What is the power of 2 in the prime factorisation of 2520?
Correct answer: B
Step 1: Write (2520=252\times10). Step 2: (252=2^2\times3^2\times7) and (10=2\times5), so (2520=2^3\times3^2\times5\times7). Step 3: The number of times 2 appears is its power.
What is the correct prime factorisation of 1575?
Correct answer: A
Step 1: Write (1575=15\times105). Step 2: (15=3\times5) and (105=3\times5\times7), so (1575=3^2\times5^2\times7). Step 3: Do not keep composite factors in the final answer.
Which number has prime factorisation (2^4\times3^2\times5)?
Correct answer: C
Step 1: Calculate (2^4=16) and (3^2=9). Step 2: (16\times9\times5=720). Step 3: To get the number from prime factorisation, multiply all factors.
If (m=2^3\times3^2\times7\times11), how many distinct prime factors does (m) have?
Correct answer: C
Step 1: Look at distinct prime factors, not powers. Step 2: In (m), the distinct primes are 2, 3, 7, and 11. Step 3: Count 2 and 3 only once each despite their powers.
If the two numbers are 27 and 64, which statement about them is correct?
Correct answer: A
Step 1: (27=3^3) and (64=2^6). Step 2: They have no common prime factor. Step 3: Therefore, they are co-prime and their HCF is 1.
If two numbers are co-prime and their product is 391, what is their LCM?
Correct answer: C
Step 1: Co-prime numbers have HCF 1. Step 2: For two numbers, product (=) HCF (\times) LCM. Step 3: Therefore, the LCM will be 391.
Which option shows only prime factorisation?
Correct answer: A
Step 1: In prime factorisation, every factor must be prime. Step 2: (2), (3), and (13) are prime, so (2^3\times3\times13) is correct. Step 3: 8, 12, and 24 are composite, so they cannot remain in the final form.
If (144=2^4\times3^2) and (168=2^3\times3\times7), what is the HCF of 144 and 168?
Correct answer: B
Step 1: The common prime factors are 2 and 3. Step 2: The smaller powers are (2^3) and (3^1). Step 3: (8\times3=24), so the HCF is 24.
If (144=2^4\times3^2) and (168=2^3\times3\times7), what is the LCM of 144 and 168?
Correct answer: B
Step 1: For LCM, take the highest powers. Step 2: The highest powers are (2^4), (3^2), and (7). Step 3: (16\times9\times7=1008), so the answer is 1008.
What is the prime factorisation of 1890?
Correct answer: A
Step 1: Write (1890=189\times10). Step 2: (189=3^3\times7) and (10=2\times5). Step 3: Therefore, (1890=2\times3^3\times5\times7).
If the prime factorisation of a number is (2^6\times3^2), what is the number?
Correct answer: B
Step 1: Calculate (2^6=64) and (3^2=9). Step 2: (64\times9=576). Step 3: In a form with powers, it is easier to evaluate powers first.
In (2^5\times3^3\times7), how many prime factors are there if repetition is counted?
Correct answer: C
Step 1: To count with repetition, add the exponents. Step 2: (2^5) gives 5, (3^3) gives 3, and (7) gives 1 factor. Step 3: Total (5+3+1=9), so the answer is 9.
If the prime factorisation of a number is (2^a\times3^2\times5) and the number is 180, what is the value of (a)?
Correct answer: B
Step 1: Prime factorise 180. Step 2: (180=18\times10=2^2\times3^2\times5). Step 3: Comparing with the given form gives (a=2).
If the prime factorisation of a number is (2^3\times3^b\times7) and the number is 1512, what is the value of (b)?
Correct answer: C
Step 1: Write (1512=8\times189). Step 2: (8=2^3) and (189=3^3\times7), so (1512=2^3\times3^3\times7). Step 3: Comparing gives (b=3).
Which statement is correct about the two numbers (2^4\times7) and (3^2\times5)?
Correct answer: A
Step 1: The prime factors of the first number are 2 and 7. Step 2: The prime factors of the second number are 3 and 5. Step 3: There is no common prime factor, so they are co-prime.
If two co-prime numbers are 28 and 45, what is their LCM?
Correct answer: D
Step 1: (28=2^2\times7) and (45=3^2\times5), so they are co-prime. Step 2: The LCM of co-prime numbers equals their product. Step 3: (28\times45=1260), so the answer is 1260.
What is the prime factorisation of 2048?
Correct answer: A
Step 1: Divide 2048 repeatedly by 2. Step 2: (2048=2^{11}). Step 3: 4 and 32 are composite, so do not use them in the final prime form.
QUIZ COMPLETE