Question

Write the sum of the powers of prime factors of 196.

Answer 1

Hint: To find the sum of the powers, we need to find the prime factors of the given number. To find the prime factors, we will divide the number by the lowest prime number. We will repeat this until the number is no longer divisible by that number. Then we will divide it with the next prime number and repeat this process until the number is completely factorised with all the factors as prime numbers. As soon as we find the prime factors, we can write them in the form of powers. Thus, we can find their sum.

Complete step-by-step answer:
The given number is 196.
We will find the factors of 196 with repeated division.
The smallest prime number is 2 and it can divide 196 completely.
Thus, 196 = 2 $\times $ 98
The quotient is 98, which is again divisible by 2.
$\Rightarrow $ 196 = 2 $\times $ 2 $\times $ 49
The quotient now is 49, which is not divisible by 2. The prime number that can divide 49 is 7.
$\Rightarrow $ 196 = 2 $\times $ 2 $\times $ 7 $\times $ 7
The quotient is 7, which is divisible by itself.
Thus, prime factorisation of 196 = 2 $\times $ 2 $\times $ 7 $\times $ 7 $\times $ 1
Now, we can also write the prime factors as 196 = ${{2}^{2}}\times {{7}^{2}}\times 1$
Therefore, power of 2 is 2 and power of 7 is 2.
Thus, the sum of the powers is 2 + 2 = 4

Note: The power of 1 is not to be considered as 1 is neither prime number nor composite number. 1 is a special case. Therefore, the sum of powers will be 4 and not 5.