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.

Practice questions

01 Which is the prime factorisation of 705600?

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Answer and explanation

Fundamental Theorem of Arithmetic

Practice questions

01 Which is the prime factorisation of 705600?

02 If (n=2^7\times3^6\times5^3\times19), by which number must (n) be divisible?

03 If (n=2^9\times3^5\times5^3), by which of the following will (n) not be divisible?

04 Which option gives the correct value of (2^{10}\times3^4\times5)?

05 If the prime factorisation of a number is (2^6\times3^6\times5^3), what is the number?

06 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)?

07 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)?

08 Which is the prime factorisation of 262440?

09 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?

10 By which smallest number should 254016 be divided to get a perfect square?

11 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)?

12 If (x=2^9\times3^5\times5) and (y=2^8\times3^2\times11^2), what will be the power of 2 in (xy)?

13 Which option gives all distinct prime factors of 9699690 correctly?

14 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)?

15 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)?

16 If the HCF of two numbers is 360 and their LCM is 50400, what is the power of 2 in their product?

17 If the HCF of two numbers is 143 and their LCM is 510510, what is the power of 13 in their product?

18 What is the prime factorisation of 160000?

19 If (n=2^{11}\times3^{10}\times5^7), by which smallest number should (n) be divided to make it a perfect cube?

20 Which option gives the number formed by (2^7\times3^4\times5\times7)?

21 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?

22 If two numbers have prime factorisations (2^7\times3^6) and (2^{10}\times3^3), what is the ratio of their LCM to HCF?

23 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?

24 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)?

25 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)?

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