Question

A chose an integer X, which is between 2 and 40. A noticed that X is such a number that, when any integer Y is divided by X, the remainder is always 1. What is the value of X?

• A. 37
• B. 41
• C. 39
• D. 41! + 1

Hint:

When solving modular arithmetic problems, always consider the largest prime number within the given range that satisfies the conditions. Primes are key to problems involving divisors and remainders

Answer 1

The Correct Option is A

Understand the Problem:

The integer \(X\) must satisfy the condition that for any integer \(Y\), dividing \(Y\) by \(X\) always gives a remainder of 1. This implies:

\[ Y \mod X = 1 \]

Characteristics of \(X\):

Since \(X\) must be between 2 and 40, we can deduce that \(X\) is not divisible by any integer between 2 and 40. In mathematical terms, \(X\) must be co-prime with all integers between 2 and 40.

Candidates for \(X\):

The largest integer that satisfies this property is a prime number below 40 that is not divisible by any number between 2 and 40. The largest prime number below 40 is 37.

Verification:

For \(X = 37\), any integer \(Y\) when divided by 37 will leave a remainder of 1:

\[ Y = k \cdot 37 + 1, \quad k \in \mathbb{Z}. \]

Thus, the value of \(X\) is 37.