Real Numbers: CBSE Class 10 Mathematics practice questions

D. Every composite number can be written as a product of primes, and this factorisation is unique apart from the order of the primes

The theorem has two parts: a composite number can be factorised into primes, and the factorisation is unique except for the order of the factors. The three rows show the same primes in different orders, so they count as one factorisation. Primes may repeat (for example $12 = 2\times2\times3$), so the last option is false.

A. $3^2\times5^2\times11$

The leaves are $5,5,3,3,11$. Grouping equal primes: $2475 = 3^2\times5^2\times11$ (check: $9\times25\times11 = 2475$). Writing each prime once gives only 165, using one 5 gives 495, and squaring 11 gives $27225$. Every exponent must match how often the prime appears.

A. 52920

$540 = 2^2\times3^3\times5$ and $1176 = 2^3\times3\times7^2$. Greatest powers: $2^3, 3^3, 5^1, 7^2$, so $\text{LCM} = 8\times27\times5\times49 = 52920$. Using $7^1$ gives 7560, the product $635040$ counts the shared factor 12 twice, and $12$ is the HCF.