Question

What can you say about the prime factorisation of the denominator of \[15.25\]\[?\]

Answer 1

Hint: Prime factorization of a given number is a process of expressing the given number as the product of prime factors. To find out the prime factorization of the denominator of \[15.25\], we have to express it in fraction form to find the denominator. Then simplify it as much as possible in the fraction form only. Then consider the denominator, expressing it as a product of prime factors.

Complete step-by-step answer:
If a number \[a\] divides another number \[b\] exactly, we say that \[a\] is a factor of \[b\]. In this case, \[b\]is called a multiple of \[a\].
A counting number is called a prime number if it has exactly two factors, namely itself and \[1\]. Examples \[2,3,5,7,11,13,17,...,etc\] are prime numbers.
Given \[15.25\]---(1)
Let the decimal value (1) be divided by \[1\], we get
\[\dfrac{{15.25}}{1}\]--(2)
Multiply the numerator and denominator of (2) with \[100\] to remove the decimal points.
\[\dfrac{{15.25 \times 100}}{{1 \times 100}} = \dfrac{{1525}}{{100}} = \dfrac{{1525}}{{100}} = \dfrac{{61}}{4}\].
So, the required denominator is \[4\].
Then the prime factorization of \[4 = 2 \times 2\]
Hence the prime factors of the denominator of \[15.25\] are \[2,2\].
Since the denominator of any number with finite decimal expansion has a prime factorization of the numbers \[2\] and (OR) \[5\] only.

Note: Note that the factorization method used to find the LCM and HCF. In the factorization method, each one of the given numbers is expressed as the product of prime factors. The product of least powers of common prime factors gives HCF. Similarly, the product of highest powers of all the factors gives LCM.