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 (N=2^{10}\times3^8\times5^5\times7), by which smallest number must (N) be multiplied to make it a perfect cube?ExpertLevel 5If (a=2^7\times3^6\times5^2) and (b=2^5\times3^8\times7^3), what will be the power of 3 in (ab)?ExpertLevel 5If (x=2^9\times3^4\times5^7) and (y=2^6\times5^5\times11^3), what will be the power of 5 in (xy)?ExpertLevel 5If (A=2^{12}\times3^8\times5^4) and (B=2^9\times3^{10}\times7^5), which is the HCF of (A) and (B)?ExpertLevel 5If (A=2^{12}\times3^8\times5^4) and (B=2^9\times3^{10}\times7^5), which is the LCM of (A) and (B)?ExpertLevel 5The HCF of two numbers is 120 and their LCM is 9240. If one number is 840, what is the other number?ExpertLevel 5The HCF of two numbers is 144 and their LCM is 10080. If one number is 720, what is the other number?ExpertLevel 5If (p=2^a\times3^5\times5\times7) and (p=136080), what is the value of (a)?ExpertLevel 5If (q=2^6\times3^b\times5^2\times7) and (q=1814400), what is the value of (b)?ExpertLevel 5If a number has prime factorisation (2^9\times3^7\times5^6\times17^2), how many prime factors are there if repetition is counted?ExpertLevel 5If a number has prime factorisation (2^{11}\times3^8\times5^3\times7^2\times13), how many distinct prime factors does it have?ExpertLevel 5If the two numbers are (2^8\times3^4\times13) and (5^5\times7^3\times11), which statement about them is correct?ExpertLevel 5If the product of two co-prime numbers is 7429, what is their LCM?ExpertLevel 5What is the prime factorisation of 65536?ExpertLevel 5Which is the prime factorisation of 705600?ExpertLevel 5If (n=2^7\times3^6\times5^3\times19), by which number must (n) be divisible?ExpertLevel 5If (n=2^9\times3^5\times5^3), by which of the following will (n) not be divisible?ExpertLevel 5Which option gives the correct value of (2^{10}\times3^4\times5)?ExpertLevel 5If the prime factorisation of a number is (2^6\times3^6\times5^3), what is the number?ExpertLevel 5If (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)?ExpertLevel 5