Factorization of Expression
We need to find the greatest prime factor of the expression $(3^{199} – 3^{196})$. First, factor out the common term, which is the lowest power of 3, $3^{196}$.
$3^{199} – 3^{196} = 3^{196} (3^{199-196} – 1)$
Simplifying the Expression
Now, simplify the term inside the parenthesis:
$3^{196} (3^3 – 1)$
Calculate $3^3$:
$3^3 = 3 \times 3 \times 3 = 27$
Substitute this back into the expression:
$3^{196} (27 – 1) = 3^{196} \times 26$
Prime Factorization
To find the greatest prime factor, we determine the prime factors of the simplified expression $3^{196} \times 26$.
- The prime factors of $3^{196}$ consist only of the prime number 3.
- Find the prime factors of 26: $26 = 2 \times 13$. Both 2 and 13 are prime numbers.
The prime factors of the entire expression $(3^{199} – 3^{196})$ are therefore 3, 2, and 13.
Greatest Prime Factor Identification
Comparing the prime factors {2, 3, 13}, the largest number is 13.
Thus, the greatest prime factor of $(3^{199} – 3^{196})$ is 13.
Important Questions from Integers
Find the number of integers between $1$ and $150$ (inclusive) having $7$ as one of the digits but which are not divisible by $7$.
Integers are listed from 700 to 1000. In how many integers is the sum of the digits 10 ?
Using 2, 2, 3, 3, 3 as digits, how many distinct numbers greater than 30000 can be formed ?
Consider the following statements :
1. The sum of 5 consecutive integers can be 100.
2 The product of three consecutive natural numbers can be equal to their sum.
Which of the above statements is/are correct ?
The difference between a 2-digit number and the number obtained by interchanging the positions of the digits is 54.
Consider the following statements:
1. The sum of the two digits of the number can be determined only if the product of the two digits is known.
2. The difference between the two digits of the number can be determined.
Which of the above statements is/are correct?
