Question

How many prime factors are there in the expression $ {(12)^{43}} \times {(34)^{48}} \times {(2)^{57}} $ ?
A. $ 282 $
B. $ 237 $
C. $ 142 $
D. $ 61 $

Answer 1

Hint: In order to solve this question first we will understand the term prime factors further we will use given expression and factorized it as \[12{\text{ }} = {\text{ }}2 \times {\text{ }}3 \times {\text{ }}2{\text{ }},{\text{ }}34{\text{ }} = {\text{ }}17 \times 2\] at last we will add power of all the prime numbers to get the required answer.

Complete step-by-step answer:
Prime factorization: Prime factorization or integer factorization of a number is breaking a number down into the set of prime numbers which multiply together to result in the original number.

Given expression is $ {(12)^{43}} \times {(34)^{48}} \times {(2)^{57}} $
Now we will break the number in to the set of prime number as
\[\begin{array}{*{20}{l}}
  {12{\text{ }} = {\text{ }}2{\text{ }} \times {\text{ }}3{\text{ }} \times {\text{ }}3} \\
  \; \\
  {34{\text{ }} = {\text{ }}17{\text{ }} \times {\text{ }}2}
\end{array}\]
Put these values in above expression we have
\[{(2 \times 2 \times 3)^{43}} \times {(2 \times 17)^{48}} \times {(2)^{57}}\]
As we know that \[{(ab)^x} = {a^x} \times {b^x}\]
By applying this property in above expression we have
\[{(2)^{43}} \times {(2)^{43}} \times {(3)^{43}} \times {(2)^{48}} \times {(17)^{48}} \times {(2)^{57}}\]
Now to get prime factors we will add power of all prime numbers
So the number of prime factors= \[43 + 43 + 43 + 48 + 48 + 57 = 282\]
Hence the correct answer is option A.

Note: Number factorization, in number theory, is the decomposition of a composite number into a set of smaller integers. The method is called prime factorization if such factors are further limited to prime numbers.