Which is the prime factorisation of 705600?
Correct answer: A
Step 1: Recognise (705600=840^2). Step 2: Since (840=2^3\times3\times5\times7), (840^2=2^6\times3^2\times5^2\times7^2). Step 3: In a perfect square, all prime exponents are even.
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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Expert · Level 4
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- (2^6\times3^2\times5^2\times7^2)
- (2^5\times3^3\times5^2\times7^2)
- (2^6\times3^2\times5^3\times7)
- (840\times840)
Expert · Level 4
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- 69255
- 23
- 29
- 31
Expert · Level 4
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- 3840
- 6075
- 16000
- 16875
Expert · Level 4
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- 414720
- 207360
- 829440
- 311040
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- 5832000
- 2916000
- 11664000
- 1944000
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- 6
- 7
- 9
- 15
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- 1
- 2
- 4
- 5
Expert · Level 4
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- (2^3\times3^8\times5)
- (2^4\times3^7\times5)
- (2^3\times3^7\times5^2)
- (52488\times5)
Expert · Level 4
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- (2\times3^2\times5^2)
- (2^2\times3\times5)
- (2\times3\times5^2)
- (2^2\times3^2\times5)
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- 1
- 2
- 3
- 7
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- 10
- 11
- 12
- 13
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- 15
- 16
- 17
- 18
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- 2, 3, 5, 7, 11, 13, 17 and 19
- 2, 3, 5, 7, 11, 13 and 17
- 3, 5, 7, 11, 13, 17 and 19
- 2, 5, 7, 11, 13, 17 and 19
Expert · Level 4
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- 323136
- 646272
- 161568
- 969408
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- 354859200
- 177429600
- 709718400
- 118286400
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- 6
- 7
- 8
- 9
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- 1
- 2
- 3
- 0
Expert · Level 4
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- (2^8\times5^4)
- (2^7\times5^5)
- (2^9\times5^3)
- (400\times400)
Expert · Level 4
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- (2^2\times3\times5)
- (2\times3^2\times5)
- (2^2\times3^2\times5)
- (2\times3\times5^2)
Expert · Level 4
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- 362880
- 181440
- 725760
- 241920
Expert · Level 4
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- (ab)
- (a+b)
- (a-b)
- 1
Expert · Level 4
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- 216
- 432
- 108
- 648
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- 34
- 35
- 36
- 37
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- 4
- 5
- 7
- 11
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- 4
- 5
- 6
- 8
If (n=2^7\times3^6\times5^3\times19), by which number must (n) be divisible?
Correct answer: A
Step 1: (69255=3^6\times5\times19). Step 2: All these prime factors are present in (n) with sufficient powers. Step 3: Therefore, (n) must be divisible by 69255.
If (n=2^9\times3^5\times5^3), by which of the following will (n) not be divisible?
Correct answer: D
Step 1: In (n), the powers are (2^9), (3^5), and (5^3). Step 2: (16875=3^3\times5^4), which needs power 4 of 5. Step 3: Since (n) has only (5^3), (n) is not divisible by 16875.
Which option gives the correct value of (2^{10}\times3^4\times5)?
Correct answer: A
Step 1: Calculate (2^{10}=1024) and (3^4=81). Step 2: (1024\times81\times5=414720). Step 3: In larger multiplication, simplify powers first.
If the prime factorisation of a number is (2^6\times3^6\times5^3), what is the number?
Correct answer: A
Step 1: Calculate (2^6=64), (3^6=729), and (5^3=125). Step 2: (64\times729\times125=5832000). Step 3: Simplifying powers first keeps the calculation clean.
If (a=2^6\times3^7\times7) and (b=2^9\times3^4\times7^5), what is the power of 2 in the LCM of (a) and (b)?
Correct answer: C
Step 1: In LCM, take the higher power of each prime. Step 2: The powers of 2 are 6 and 9. Step 3: The higher power is 9, so the answer is 9.
If (a=2^6\times3^7\times7) and (b=2^9\times3^4\times7^5), what is the power of 7 in the HCF of (a) and (b)?
Correct answer: A
Step 1: In HCF, take the smaller power of the common prime. Step 2: The powers of 7 are 1 and 5. Step 3: The smaller power is 1, so the answer is 1.
Which is the prime factorisation of 262440?
Correct answer: A
Step 1: Write (262440=52488\times5). Step 2: (52488=8\times6561=2^3\times3^8). Step 3: Therefore, (262440=2^3\times3^8\times5).
If a number has prime factorisation (2^{11}\times3^7\times5^4), what is the smallest number by which it must be multiplied to make a perfect cube?
Correct answer: A
Step 1: For a perfect cube, exponents must be multiples of 3. Step 2: Make (2^{11}), (3^7), and (5^4) into powers 12, 9, and 6. Step 3: The smallest multiplier is (2\times3^2\times5^2).
By which smallest number should 254016 be divided to get a perfect square?
Correct answer: A
Step 1: (254016=64\times3969=2^6\times3^4\times7^2). Step 2: All exponents are even, so the number is already a perfect square. Step 3: If it is already a perfect square, the smallest divisor is 1.
If (x=2^7\times3^4\times7^5) and (y=2^5\times5^3\times7^6), what will be the power of 7 in (xy)?
Correct answer: B
Step 1: In multiplication, exponents of the same prime base are added. Step 2: The power of 7 in (x) is 5 and in (y) is 6. Step 3: In (xy), the power of 7 is (5+6=11).
If (x=2^9\times3^5\times5) and (y=2^8\times3^2\times11^2), what will be the power of 2 in (xy)?
Correct answer: C
Step 1: Powers of the same base 2 are added in multiplication. Step 2: The power of 2 in (x) is 9 and in (y) is 8. Step 3: The total power is (9+8=17).
Which option gives all distinct prime factors of 9699690 correctly?
Correct answer: A
Step 1: Write (9699690=510510\times19). Step 2: (510510=2\times3\times5\times7\times11\times13\times17). Step 3: Therefore, the distinct prime factors are 2, 3, 5, 7, 11, 13, 17, and 19.
If (a=2^7\times3^4\times13^2) and (b=2^5\times3^6\times5^2\times13), what is the HCF of (a) and (b)?
Correct answer: A
Step 1: The common prime factors are 2, 3, and 13. Step 2: The smaller powers are (2^5), (3^4), and (13^1). Step 3: (32\times81\times13=33696), so the correct HCF is 33696.
If (a=2^7\times3^4\times13^2) and (b=2^5\times3^6\times5^2\times13), what is the LCM of (a) and (b)?
Correct answer: A
Step 1: For LCM, take the highest powers of all prime factors. Step 2: The highest powers are (2^7), (3^6), (5^2), and (13^2). Step 3: (128\times729\times25\times169=394221600), so the correct value is 394221600.
If the HCF of two numbers is 360 and their LCM is 50400, what is the power of 2 in their product?
Correct answer: B
Step 1: The product is (360\times50400). Step 2: (360=2^3\times3^2\times5) and (50400=2^5\times3^2\times5^2\times7). Step 3: The power of 2 in the product is (3+5=8).
If the HCF of two numbers is 143 and their LCM is 510510, what is the power of 13 in their product?
Correct answer: B
Step 1: The product is (143\times510510). Step 2: (143=11\times13), and 510510 has 13 to power 1. Step 3: Therefore, the power of 13 in the product is (1+1=2).
What is the prime factorisation of 160000?
Correct answer: A
Step 1: Write (160000=16\times10000). Step 2: (16=2^4) and (10000=10^4=2^4\times5^4). Step 3: Therefore, (160000=2^8\times5^4).
If (n=2^{11}\times3^{10}\times5^7), by which smallest number should (n) be divided to make it a perfect cube?
Correct answer: A
Step 1: In a perfect cube, every exponent must be a multiple of 3. Step 2: Reduce (2^{11}) to (2^9), (3^{10}) to (3^9), and (5^7) to (5^6). Step 3: So the smallest divisor is (2^2\times3\times5).
Which option gives the number formed by (2^7\times3^4\times5\times7)?
Correct answer: A
Step 1: Calculate (2^7=128) and (3^4=81). Step 2: (128\times81\times5\times7=362880). Step 3: To get the number from prime factorisation, multiply all factors.
If (a=2^8\times3^4\times5) and (b=2^5\times3^7\times5^2), the product of their HCF and LCM will be equal to what?
Correct answer: A
Step 1: For two numbers, HCF (\times) LCM equals the product of the two numbers. Step 2: In prime powers, the smaller and higher exponents together give the total exponent. Step 3: Therefore, the answer is (ab).
If two numbers have prime factorisations (2^7\times3^6) and (2^{10}\times3^3), what is the ratio of their LCM to HCF?
Correct answer: A
Step 1: HCF is (2^7\times3^3). Step 2: LCM is (2^{10}\times3^6). Step 3: The ratio is (2^3\times3^3=8\times27=216).
If (x=2^6\times3^7\times5^5) and (y=2^8\times3^4\times5^6), how many prime factors will (xy) have if repetition is counted?
Correct answer: C
Step 1: In (xy), exponents of the same bases are added. Step 2: Powers become (2^{14}), (3^{11}), and (5^{11}). Step 3: Counting with repetition gives (14+11+11=36).
If (m=2^9\times3^5\times5^4) and (n=2^6\times3^8\times5^7), what is the power of 5 in the LCM of (m) and (n)?
Correct answer: C
Step 1: In LCM, take the higher power. Step 2: The powers of 5 are 4 and 7. Step 3: The higher power is 7, so the answer is 7.
If (m=2^9\times3^5\times5^4) and (n=2^6\times3^8\times5^7), what is the power of 3 in the HCF of (m) and (n)?
Correct answer: B
Step 1: In HCF, take the smaller power. Step 2: The powers of 3 are 5 and 8. Step 3: The smaller power is 5, so the answer is 5.
QUIZ COMPLETE