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
If the HCF of two numbers is 78 and their LCM is 510510, what is the power of 13 in their product?ExpertLevel 4What is the prime factorisation of 83521?ExpertLevel 4If (n=2^{10}\times3^8\times5^6), by which smallest number should (n) be divided to make it a perfect cube?ExpertLevel 4Which option gives the number formed by (2^6\times3^4\times5\times7)?ExpertLevel 4If (a=2^7\times3^3\times5) and (b=2^4\times3^6\times5^2), the product of their HCF and LCM will be equal to what?ExpertLevel 4If two numbers have prime factorisations (2^6\times3^5) and (2^9\times3^2), what is the ratio of their LCM to HCF?ExpertLevel 4If (x=2^5\times3^6\times5^4) and (y=2^7\times3^3\times5^5), how many prime factors will (xy) have if repetition is counted?ExpertLevel 4If (m=2^8\times3^4\times5^3) and (n=2^5\times3^7\times5^6), what is the power of 5 in the LCM of (m) and (n)?ExpertLevel 4If (m=2^8\times3^4\times5^3) and (n=2^5\times3^7\times5^6), what is the power of 3 in the HCF of (m) and (n)?ExpertLevel 4According to the Fundamental Theorem of Arithmetic, up to what extent is the prime factorisation of a number unique?ExpertLevel 5Which is the correct prime factorisation of 221760?ExpertLevel 5If (a=2^8\times3^5\times5^2) and (b=2^5\times3^7\times7^3), what is the HCF of (a) and (b)?ExpertLevel 5If (a=2^8\times3^5\times5^2) and (b=2^5\times3^7\times7^3), what is the LCM of (a) and (b)?ExpertLevel 5If the product of two numbers is 1814400 and their HCF is 720, what is their LCM?ExpertLevel 5If the HCF of two numbers is 210 and their LCM is 9240, what is their product?ExpertLevel 5What is the smallest positive number by which 21600 must be multiplied to get a perfect square?ExpertLevel 5By which smallest number should 52920 be divided to get a perfect square?ExpertLevel 5What is the smallest number by which 30240 must be multiplied to get a perfect cube?ExpertLevel 5What is the prime factorisation of 500094?ExpertLevel 5If (N=2^{11}\times3^8\times5^7\times13^2), by which smallest number must (N) be multiplied to make it a perfect square?ExpertLevel 5