The number of pears at a fruit stall is between 49 to 75. The stallholder will have a remainder of 2 pears whether he sells the pears at 3 for $1 or at 5 for $1. How many pears are there at the stall?

Let the number of pears be denoted by $ n $. According to the problem, $ n $ gives a remainder of 2 when divided by both 3 and 5. This can be expressed mathematically as:

\[
n \equiv 2 \pmod{3}
\]
\[
n \equiv 2 \pmod{5}
\]

Since $ n \equiv 2 \pmod{3} $ and $ n \equiv 2 \pmod{5} $, $ n $ must satisfy the following congruences:

\[
n = 3k + 2 \quad \text{for some integer } k
\]
\[
n = 5m + 2 \quad \text{for some integer } m
\]

Since both expressions for $ n $ have the same remainder 2, we equate them:

\[
3k + 2 = 5m + 2
\]

Subtracting 2 from both sides:

\[
3k = 5m
\]

This implies that $ 3k $ is a multiple of 5. The smallest $ k $ that makes $ 3k $ a multiple of 5 is when $ k $ is a multiple of 5. Let $ k = 5p $, where $ p $ is an integer.

Thus:

\[
n = 3(5p) + 2 = 15p + 2
\]

Now, $ n = 15p + 2 $ must be within the range 49 to 75. We solve for $ p $ by setting up inequalities:

\[
49 \leq 15p + 2 \leq 75
\]

Subtract 2 from all parts of the inequality:

\[
47 \leq 15p \leq 73
\]

Divide by 15:

\[
\frac{47}{15} \leq p \leq \frac{73}{15}
\]

This simplifies to:

\[
3.13 \leq p \leq 4.87
\]

Since $ p $ must be an integer, the only possible value for $ p $ is 4.

Substitute $ p = 4 $ back into the expression for $ n $:

\[
n = 15(4) + 2 = 60 + 2 = 62
\]

Thus, the number of pears at the stall is **62**.

Would you like more details, or have any other questions?

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Here are 5 related questions for further exploration:

1. What if the number of pears gave a remainder of 1 instead of 2 when divided by 3 and 5?
2. How would the solution change if the range of pears was different, say between 30 and 90?
3. Can this method be applied to other numbers or different moduli?
4. How can this problem be generalized for any two divisors?
5. What would happen if there were additional conditions, such as a different remainder when divided by another number?

**Tip:** Understanding modular arithmetic is essential for solving problems involving remainders and divisibility.

Explore the solution to a modular arithmetic problem involving the number of pears at a fruit stall, where the remainder is 2 when divided by both 3 and 5. Understand the application of modular arithmetic and linear Diophantine equations. Suitable for advanced high school students and mathematics enthusiasts.

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