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

To solve the problem, we need to find an integer \( X \) between 2 and 40 such that when any integer \( Y \) is divided by \( X \), the remainder is always 1. This implies that \( X \) divides \( Y - 1 \). Therefore, \( X \) must be a number which only leaves a remainder of 1 when dividing such that \( Y = nX + 1 \) for any integer \( n \).

To clarify, when \( Y \) is divided by \( X \), the expression \( Y \mod X = 1 \) holds true. Thus, \( X \) must satisfy the condition \( (nX + 1) \mod X = 1 \), which simplifies to the condition that \( X > 1 \). 

Now, let's analyze the options:

1. Option 1: 37. If \( X = 37 \), for any \( Y \), the expression \( Y - 1 \) must be divisible by 37. Since 37 is a prime, it can divide any number greater than itself minus one, hence any integer remainder condition \( Y = 37k + 1 \) holds.
2. Option 2: 41. This number is not between 2 and 40.
3. Option 3: 39. It is not a prime number, and generally cannot always produce remainder 1 except if \( Y-1 \) is specifically created to be a multiple of 39.
4. Option 4: 41! + 1. This number is much larger than 40 and exceeds the range given.

Based on the analysis above, the integer \( X \) that satisfies all the conditions is 37. Therefore, the correct answer is:

37

Answer 2

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.