If (n=2^2\times3^4\times5), what is the value of (n)?
Correct answer: C
Step 1: Calculate (2^2=4) and (3^4=81). Step 2: (4\times81\times5=1620). Step 3: Simplifying powers first makes multiplication easier.
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 4
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- 810
- 1080
- 1620
- 3240
Medium · Level 4
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- (3^2\times5^3)
- (3^3\times5^2)
- (3\times5^4)
- (45\times25)
Medium · Level 4
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- 3
- 5
- 7
- 9
Medium · Level 4
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- 108
- 216
- 324
- 972
Medium · Level 4
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- 1296
- 3888
- 7776
- 15552
Medium · Level 4
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- 3, 5, 7 and 13
- 3, 5, 7 and 11
- 5, 7, 11 and 13
- 3, 5 and 13
Medium · Level 4
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- 448
- 11760
- 392
- 15
Medium · Level 4
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- 12
- 14
- 18
- 24
Medium · Level 4
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- (3^3\times5^3)
- (3^2\times5^4)
- (3^4\times5^2)
- (27\times125)
Medium · Level 4
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- 1 is neither prime nor composite
- 1 is prime because it divides all numbers
- 1 is composite because it is small
- 1 has no positive factor
Medium · Level 4
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- 12
- 24
- 36
- 60
Medium · Level 4
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- 8820
- 17640
- 35280
- 4410
Medium · Level 4
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- (2^2\times3^2\times7\times11)
- (2^3\times3\times7\times11)
- (2^2\times3\times7^2\times11)
- (252\times11)
Medium · Level 4
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- Every number greater than 1 has a unique prime factorisation except for order
- Every number has HCF 1
- Every number is co-prime
- Every prime factorisation has only two factors
Medium · Level 4
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- 3600
- 7200
- 14400
- 18000
Medium · Level 4
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- Co-prime numbers
- Composite numbers
- Even numbers
- Perfect square numbers
Medium · Level 4
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- 336
- 672
- 1008
- 1344
Medium · Level 4
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- 2
- 3
- 4
- 5
Medium · Level 4
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- 1
- 2
- 3
- 4
Medium · Level 4
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- 2
- 3
- 4
- 5
Medium · Level 4
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- 30
- 7
- 11
- 9
Medium · Level 4
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- (3^2\times7^3)
- (3^3\times7^2)
- (3\times7^4)
- (63\times49)
Medium · Level 4
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- 5
- 6
- 8
- 9
Medium · Level 4
View options
- 360
- 720
- 1080
- 1440
Which option gives the correct prime factorisation of 1125?
Correct answer: A
Step 1: Write (1125=45\times25). Step 2: (45=3^2\times5) and (25=5^2), so (1125=3^2\times5^3). Step 3: 45 and 25 are composite, so write their prime powers in the final form.
If a number has prime factorisation (2^3\times3^2\times7^2), how many prime factors does it have including repetition?
Correct answer: C
Step 1: To count with repetition, add the exponents. Step 2: (2^3) gives 3, (3^2) gives 2, and (7^2) gives 2 factors. Step 3: Total (3+2+2=7), so the answer is 7.
If (A=2^4\times3^3) and (B=2^2\times3^5), what is the HCF of (A) and (B)?
Correct answer: A
Step 1: For HCF, take smaller powers. Step 2: The smaller power of 2 is 2 and of 3 is 3. Step 3: (2^2\times3^3=4\times27=108), so the answer is 108.
If (A=2^4\times3^3) and (B=2^2\times3^5), what is the LCM of (A) and (B)?
Correct answer: B
Step 1: For LCM, take the highest powers. Step 2: The highest powers are (2^4) and (3^5). Step 3: (16\times243=3888), so the answer is 3888.
What are the distinct prime factors of 1365?
Correct answer: A
Step 1: Write (1365=105\times13). Step 2: (105=3\times5\times7), so (1365=3\times5\times7\times13). Step 3: The distinct prime factors are 3, 5, 7, and 13.
If the HCF of two numbers is 28 and their LCM is 420, what is their product?
Correct answer: B
Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: (28\times420=11760). Step 3: In this relation, multiply the two given values directly.
If the product of two numbers is 5040 and their LCM is 360, what is their HCF?
Correct answer: B
Step 1: For two numbers, product (=) HCF (\times) LCM. Step 2: HCF (=5040\div360=14). Step 3: After finding the answer, you can check by multiplying.
What is the prime factorisation of 3375?
Correct answer: A
Step 1: Write (3375=27\times125). Step 2: (27=3^3) and (125=5^3), so (3375=3^3\times5^3). Step 3: 27 and 125 are composite, so keep prime powers in the final form.
Which option gives the correct understanding about 1?
Correct answer: A
Step 1: A prime number must have exactly two positive factors. Step 2: 1 has only one positive factor, so it is not prime. Step 3: A composite number needs more than two factors, so 1 is not composite either.
If (p=2^3\times3^2\times5) and (q=2^2\times3\times7^2), what is the HCF of (p) and (q)?
Correct answer: A
Step 1: The common prime factors are 2 and 3. Step 2: The smaller powers are (2^2) and (3^1). Step 3: (2^2\times3=4\times3=12), so the HCF is 12.
If (p=2^3\times3^2\times5) and (q=2^2\times3\times7^2), what is the LCM of (p) and (q)?
Correct answer: B
Step 1: For LCM, take the highest powers. Step 2: The highest powers are (2^3), (3^2), (5), and (7^2). Step 3: (8\times9\times5\times49=17640), so the answer is 17640.
Which is the prime factorisation of 2772?
Correct answer: A
Step 1: Write (2772=252\times11). Step 2: (252=2^2\times3^2\times7), so (2772=2^2\times3^2\times7\times11). Step 3: 252 is composite, so it should not remain in the final form.
Which conclusion correctly follows from the Fundamental Theorem of Arithmetic?
Correct answer: A
Step 1: The theorem is about prime factorisation of numbers greater than 1. Step 2: This factorisation is unique except for order. Step 3: Do not mix it with separate rules of HCF or co-primality.
If (r=2^5\times3^2\times5^2), what is the value of (r)?
Correct answer: B
Step 1: Calculate (2^5=32), (3^2=9), and (5^2=25). Step 2: (32\times9\times25=7200). Step 3: In larger multiplication, simplify powers first.
If two numbers have no common prime factor in their prime factorisations, what are they called?
Correct answer: A
Step 1: Having no common prime factor means their only common factor is 1. Step 2: Such numbers are called co-prime numbers. Step 3: Prime factorisation helps identify co-primality quickly.
Which number has prime factorisation (2^5\times3\times7)?
Correct answer: B
Step 1: Calculate (2^5=32). Step 2: (32\times3\times7=672). Step 3: To get the number from prime factorisation, multiply all factors.
If the prime factorisation of a number is (2^4\times3^a\times5) and the number is 6480, what is the value of (a)?
Correct answer: C
Step 1: Write (6480=648\times10). Step 2: (648=2^3\times3^4) and (10=2\times5), so (6480=2^4\times3^4\times5). Step 3: Comparing gives (a=4).
If (a=2^3\times3^2\times5) and (b=2^2\times3\times5^2), what will be the power of 5 in the prime factorisation of (ab)?
Correct answer: C
Step 1: While multiplying, exponents of the same prime base are added. Step 2: The power of 5 in (a) is 1 and in (b) is 2. Step 3: In (ab), the power of 5 will be (1+2=3).
If (a=2^4\times7^2) and (b=2^3\times7), what will be the power of 7 in the prime factorisation of (ab)?
Correct answer: B
Step 1: Powers with the same base are added in multiplication. Step 2: The power of 7 in (a) is 2 and in (b) is 1. Step 3: In (ab), the power of 7 will be (2+1=3).
If a number has (2^2), (3), and (5) in its prime factorisation, by which number must it be divisible?
Correct answer: A
Step 1: The factorisation contains 2, 3, and 5. Step 2: (2\times3\times5=30), so the number must be divisible by 30. Step 3: Identify divisibility quickly from prime factors.
Which is the prime factorisation of 3087?
Correct answer: A
Step 1: Write (3087=63\times49). Step 2: (63=3^2\times7) and (49=7^2). Step 3: Therefore, (3087=3^2\times7^3).
If (x=2^5\times3^2) and (y=2^3\times3^4), what will be the power of 2 in (xy)?
Correct answer: C
Step 1: In (xy), the powers of the same base 2 are added. Step 2: The power of 2 in (x) is 5 and in (y) is 3. Step 3: So the power of 2 in (xy) is (5+3=8).
If (x=2^5\times3^2\times5) and (y=2^3\times3^4\times5^2), what is the HCF of (x) and (y)?
Correct answer: A
Step 1: For HCF, take the smaller powers of common prime factors. Step 2: The smaller powers are (2^3), (3^2), and (5^1). Step 3: (8\times9\times5=360), so the HCF is 360.
QUIZ COMPLETE