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 (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)?ExpertLevel 5Which is the prime factorisation of 262440?ExpertLevel 5If 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?ExpertLevel 5By which smallest number should 254016 be divided to get a perfect square?ExpertLevel 5If (x=2^7\times3^4\times7^5) and (y=2^5\times5^3\times7^6), what will be the power of 7 in (xy)?ExpertLevel 5If (x=2^9\times3^5\times5) and (y=2^8\times3^2\times11^2), what will be the power of 2 in (xy)?ExpertLevel 5Which option gives all distinct prime factors of 9699690 correctly?ExpertLevel 5If (a=2^7\times3^4\times13^2) and (b=2^5\times3^6\times5^2\times13), what is the HCF of (a) and (b)?ExpertLevel 5If (a=2^7\times3^4\times13^2) and (b=2^5\times3^6\times5^2\times13), what is the LCM of (a) and (b)?ExpertLevel 5If the HCF of two numbers is 360 and their LCM is 50400, what is the power of 2 in their product?ExpertLevel 5If the HCF of two numbers is 143 and their LCM is 510510, what is the power of 13 in their product?ExpertLevel 5What is the prime factorisation of 160000?ExpertLevel 5If (n=2^{11}\times3^{10}\times5^7), by which smallest number should (n) be divided to make it a perfect cube?ExpertLevel 5Which option gives the number formed by (2^7\times3^4\times5\times7)?ExpertLevel 5If (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?ExpertLevel 5If two numbers have prime factorisations (2^7\times3^6) and (2^{10}\times3^3), what is the ratio of their LCM to HCF?ExpertLevel 5If (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?ExpertLevel 5If (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)?ExpertLevel 5If (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)?ExpertLevel 5According to the Fundamental Theorem of Arithmetic, what is the correct meaning of certainty in prime factorisation?ExpertLevel 6