The integers 34041 and 32506, when divided by a 3 - digit integer n, leave the same remainder. What can be the value of n?
The problem is based on a fundamental property of modular arithmetic. If two integers, 'a' and 'b', leave the same remainder when divided by an integer 'n', then their difference (a - b) must be perfectly divisible by 'n'.
In this case, a = 34041, b = 32506, and 'n' is the 3-digit integer divisor.
According to the principle, the difference between the two numbers must be a multiple of 'n'.
Difference = 34041 - 32506 = 1535
This means that 'n' must be a factor of 1535.
To find the factors of 1535, we can perform prime factorization:
We then check if 307 is a prime number. We only need to test for divisibility by prime numbers up to $\sqrt{307}$ (which is approximately 17.5). Since 307 is not divisible by 2, 3, 5, 7, 11, 13, or 17, it is a prime number.
Therefore, the prime factorization of 1535 is 5 × 307.
The factors of 1535 are 1, 5, 307, and 1535.
The problem states that 'n' is a 3-digit integer. From the list of factors {1, 5, 307, 1535}, the only number that has three digits is 307.
The value of n must be 307.
Reinforce your conceptual grasp with related questions from NIMCET 2019:
Take timed full-length mock tests with authentic question distributions, negative marking (+12/-3 in Math), and instant All India Rank calculation.
Start Free Mock Test Series →