Which is the prime factorisation of 2822400?
Correct answer: A
Step 1: Recognise (2822400=1680^2). Step 2: Since (1680=2^4\times3\times5\times7), (1680^2=2^8\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
Quiz this set
Up to 25 questions from this page. Select your focus, then start.
25 questions
Choose questions
Expert · Level 6
View options
- (2^8\times3^2\times5^2\times7^2)
- (2^7\times3^3\times5^2\times7^2)
- (2^8\times3^2\times5^3\times7)
- (1680\times1680)
Expert · Level 6
View options
- 502929
- 29
- 31
- 37
Expert · Level 6
View options
- 12800
- 18225
- 80000
- 28125
Expert · Level 6
View options
- 2488320
- 1244160
- 4976640
- 1866240
Expert · Level 6
View options
- 11664000
- 5832000
- 23328000
- 3888000
Expert · Level 6
View options
- 7
- 8
- 10
- 17
Expert · Level 6
View options
- 1
- 3
- 5
- 6
Expert · Level 6
View options
- (2^4\times3^8\times5)
- (2^5\times3^7\times5)
- (2^4\times3^7\times5^2)
- (104976\times5)
Expert · Level 6
View options
- (3\times5)
- (2\times3\times5)
- (3^2\times5)
- (2^2\times3\times5^2)
Expert · Level 6
View options
- 1
- 2
- 3
- 7
Expert · Level 6
View options
- 12
- 13
- 14
- 15
Expert · Level 6
View options
- 17
- 18
- 19
- 20
Expert · Level 6
View options
- 2, 3, 5, 7, 11, 13, 17, 19 and 23
- 2, 3, 5, 7, 11, 13, 17 and 19
- 3, 5, 7, 11, 13, 17, 19 and 23
- 2, 5, 7, 11, 13, 17, 19 and 23
Expert · Level 6
View options
- 264384
- 132192
- 528768
- 793152
Expert · Level 6
View options
- (2^8\times3^7\times5^2\times17^2)
- (2^6\times3^5\times17)
- (2^8\times3^5\times17^2)
- (2^6\times3^7\times5^2\times17)
Expert · Level 6
View options
- 8
- 9
- 10
- 11
Expert · Level 6
View options
- 1
- 2
- 3
- 0
Expert · Level 6
View options
- (2^9\times5^4)
- (2^8\times5^5)
- (2^{10}\times5^3)
- (800\times400)
Expert · Level 6
View options
- (3^2\times5^2)
- (2\times3^2\times5)
- (3\times5^2)
- (2^2\times3\times5)
Expert · Level 6
View options
- 725760
- 362880
- 1451520
- 483840
Expert · Level 6
View options
- (ab)
- (a+b)
- (a-b)
- 1
Expert · Level 6
View options
- 216
- 432
- 108
- 648
Expert · Level 6
View options
- 40
- 41
- 42
- 43
Expert · Level 6
View options
- 5
- 6
- 8
- 13
Expert · Level 6
View options
- 5
- 6
- 7
- 9
If (n=2^8\times3^7\times5^4\times23), by which number must (n) be divisible?
Correct answer: A
Step 1: (502929=3^7\times23). Step 2: Both prime factors are present in (n) with sufficient powers. Step 3: Therefore, (n) must be divisible by 502929.
If (n=2^{10}\times3^6\times5^4), by which of the following will (n) not be divisible?
Correct answer: D
Step 1: In (n), the powers are (2^{10}), (3^6), and (5^4). Step 2: (28125=3^2\times5^5), which needs power 5 of 5. Step 3: Since (n) has only (5^4), (n) is not divisible by 28125.
Which option gives the correct value of (2^{11}\times3^5\times5)?
Correct answer: A
Step 1: Calculate (2^{11}=2048) and (3^5=243). Step 2: (2048\times243\times5=2488320). Step 3: In larger multiplication, simplify powers first.
If the prime factorisation of a number is (2^7\times3^6\times5^3), what is the number?
Correct answer: A
Step 1: Calculate (2^7=128), (3^6=729), and (5^3=125). Step 2: (128\times729\times125=11664000). Step 3: Simplifying powers first keeps the calculation clear.
If (a=2^7\times3^8\times7) and (b=2^{10}\times3^5\times7^6), 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 7 and 10. Step 3: The higher power is 10, so the answer is 10.
If (a=2^7\times3^8\times7) and (b=2^{10}\times3^5\times7^6), 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 6. Step 3: The smaller power is 1, so the answer is 1.
Which is the prime factorisation of 524880?
Correct answer: A
Step 1: Write (524880=104976\times5). Step 2: (104976=16\times6561=2^4\times3^8). Step 3: Therefore, (524880=2^4\times3^8\times5).
If a number has prime factorisation (2^{12}\times3^8\times5^5), 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: (2^{12}) is already suitable, while (3^8) and (5^5) must become (3^9) and (5^6). Step 3: The smallest multiplier is (3\times5).
By which smallest number should 254016 be divided to get a perfect square?
Correct answer: A
Step 1: (254016=2^6\times3^4\times7^2). Step 2: All powers 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=2^8\times3^5\times7^6) and (y=2^6\times5^4\times7^7), 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 6 and in (y) is 7. Step 3: In (xy), the power of 7 is (6+7=13).
If (x=2^{10}\times3^6\times5) and (y=2^9\times3^3\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 10 and in (y) is 9. Step 3: The total power is (10+9=19).
Which option gives all distinct prime factors of 223092870 correctly?
Correct answer: A
Step 1: Write (223092870=9699690\times23). Step 2: (9699690=2\times3\times5\times7\times11\times13\times17\times19). Step 3: Therefore, the distinct prime factors are 2, 3, 5, 7, 11, 13, 17, 19, and 23.
If (a=2^8\times3^5\times17^2) and (b=2^6\times3^7\times5^2\times17), what is the HCF of (a) and (b)?
Correct answer: A
Step 1: The common prime factors are 2, 3, and 17. Step 2: The smaller powers are (2^6), (3^5), and (17^1). Step 3: (64\times243\times17=264384), so the HCF is 264384.
If (a=2^8\times3^5\times17^2) and (b=2^6\times3^7\times5^2\times17), which 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 (2^8), (3^7), (5^2), and (17^2). Step 3: So the correct form is (2^8\times3^7\times5^2\times17^2).
If the HCF of two numbers is 720 and their LCM is 100800, what is the power of 2 in their product?
Correct answer: C
Step 1: The product is (720\times100800). Step 2: (720=2^4\times3^2\times5) and (100800=2^6\times3^2\times5^2\times7). Step 3: The power of 2 in the product is (4+6=10).
If the HCF of two numbers is 221 and their LCM is 9699690, what is the power of 17 in their product?
Correct answer: B
Step 1: The product is (221\times9699690). Step 2: (221=13\times17), and 9699690 has 17 to power 1. Step 3: Therefore, the power of 17 in the product is (1+1=2).
What is the prime factorisation of 320000?
Correct answer: A
Step 1: Write (320000=32\times10000). Step 2: (32=2^5) and (10000=10^4=2^4\times5^4). Step 3: Therefore, (320000=2^9\times5^4).
If (n=2^{12}\times3^{11}\times5^8), 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: (2^{12}) is suitable, while (3^{11}) and (5^8) must be reduced to (3^9) and (5^6). Step 3: So the smallest divisor is (3^2\times5^2).
Which option gives the number formed by (2^8\times3^4\times5\times7)?
Correct answer: A
Step 1: Calculate (2^8=256) and (3^4=81). Step 2: (256\times81\times5\times7=725760). Step 3: To get the number from prime factorisation, multiply all factors.
If (a=2^9\times3^5\times5) and (b=2^6\times3^8\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 (2^8\times3^8) and (2^{11}\times3^5), what is the ratio of their LCM to HCF?
Correct answer: A
Step 1: HCF is (2^8\times3^5). Step 2: LCM is (2^{11}\times3^8). Step 3: The ratio is (2^3\times3^3=8\times27=216).
If (x=2^7\times3^8\times5^6) and (y=2^9\times3^5\times5^7), 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^{16}), (3^{13}), and (5^{13}). Step 3: Counting with repetition gives (16+13+13=42).
If (m=2^{10}\times3^6\times5^5) and (n=2^7\times3^9\times5^8), 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 5 and 8. Step 3: The higher power is 8, so the answer is 8.
If (m=2^{10}\times3^6\times5^5) and (n=2^7\times3^9\times5^8), 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 6 and 9. Step 3: The smaller power is 6, so the answer is 6.
QUIZ COMPLETE