Understanding the Least Integers with Remainder 1 when Divided by Integers 2 to 11

In this article, we delve into the mathematical problem of finding the smallest integers that leave a remainder of 1 when divided by every integer from 2 to 11. This exploration will help us understand the underlying principles and methods used to solve such problems in number theory.

Problem Statement and Methodology

The problem at hand is to find the least integers greater than 1 that, when divided by any integer k where 2 leq k leq 11, leave a remainder of 1. Such integers can be represented as:

n equiv 1 pmod{k}

This condition implies that n - 1 is divisible by k. Therefore, we can express n - 1 as:

n - 1 m cdot k

where m is an integer. To satisfy the condition for all k from 2 to 11, n - 1 must be a common multiple of the numbers from 2 to 11. The least common multiple (LCM) of these integers will give us the smallest value of n - 1.

Calculating the LCM of Integers from 2 to 11

We will calculate the LCM of the integers 2 through 11 by determining their prime factorizations and then taking the highest powers of each prime factor:

2 2^1 3 3^1 4 2^2 5 5^1 6 2^1 cdot 3^1 7 7^1 8 2^3 9 3^2 10 2^1 cdot 5^1 11 11^1

Taking the highest power of each prime factor, we find:

2: 2^3 8 3: 3^2 9 5: 5^1 5 7: 7^1 7 11: 11^1 11

The LCM is the product of these prime powers:

LCM 2^3 cdot 3^2 cdot 5^1 cdot 7^1 cdot 11^1 27720

Finding the Least Integers n

Since n - 1 27720, the smallest integer n is:

n 27720 1 27721

The next integer n that satisfies the condition is:

n 27720 cdot 2 1 55441

Calculating the Difference

The difference between these two least integers is:

55441 - 27721 27720

Conclusion

The difference between the two least integers greater than 1 that leave a remainder of 1 when divided by every integer from 2 to 11 is:

27720

This solution is equivalent to finding the remainders when divided by the least multiple of the numbers 2 through 11, which is 27720. The smallest natural numbers that leave a remainder of 1 modulo 27720 are 1 and 27721, and their difference is 27720 itself.

Key Terms: least integers, remainder, LCM, mathematical problem