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
Which is the prime factorisation of 55440?HardLevel 5If (a=2^5\times3^4\times5^2) and (b=2^3\times3^6\times7^2), what is the HCF of (a) and (b)?HardLevel 5If (a=2^5\times3^4\times5^2) and (b=2^3\times3^6\times7^2), what is the LCM of (a) and (b)?HardLevel 5If the product of two numbers is 181440 and their HCF is 144, what is their LCM?HardLevel 5If the HCF of two numbers is 84 and their LCM is 1980, what is their product?HardLevel 5What is the smallest positive number by which 3528 must be multiplied to get a perfect square?HardLevel 5By which smallest number should 13230 be divided to get a perfect square?HardLevel 5What is the smallest number by which 5400 must be multiplied to get a perfect cube?HardLevel 5What is the prime factorisation of 74088?HardLevel 5If (N=2^7\times3^4\times5^3\times11), by which smallest number must (N) be multiplied to make it a perfect square?HardLevel 5If (N=2^5\times3^7\times5^2), by which smallest number must (N) be multiplied to make it a perfect cube?HardLevel 5If (a=2^4\times3^3\times7) and (b=2^3\times3^5\times5), what will be the power of 3 in (ab)?HardLevel 5If (x=2^8\times3^2\times5^3) and (y=2^3\times3^4\times5^2), what will be the power of 5 in (xy)?HardLevel 5If (A=2^9\times3^5\times5^2) and (B=2^6\times3^7\times11), which is the HCF of (A) and (B)?HardLevel 5If (A=2^9\times3^5\times5^2) and (B=2^6\times3^7\times11), which is the LCM of (A) and (B)?HardLevel 5The HCF of two numbers is 36 and their LCM is 1260. If one number is 180, what is the other number?HardLevel 5The HCF of two numbers is 42 and their LCM is 1386. If one number is 198, what is the other number?HardLevel 5If (p=2^a\times3^3\times5^2) and (p=5400), what is the value of (a)?HardLevel 5If (q=2^4\times3^b\times7^2) and (q=15876), what is the value of (b)?HardLevel 5If a number has prime factorisation (2^6\times3^4\times5^3\times7), how many prime factors are there if repetition is counted?HardLevel 5