A computer generates a series of five-digit product codes according to the following rules: The codes use the numerals 0,1,2,3,4 and no others. Each numeral occurs exactly once in any code. The second digit in the code has a value exactly twice of the first digit. The value of the third digit in the code is less than the value of the fifth digit. If the fifth digit of an acceptable product code is 1 then which of the following must be true? (1) first digit is 2 (2) second digit is 0 (3) third digit is 3 (4) fourth digit is 4 (5) fourth digit is 0

Let's analyze the problem step by step, considering the rules provided for the five-digit product codes:

1. **The codes use the numerals 0, 1, 2, 3, 4 and no others. Each numeral occurs exactly once in any code.**
2. **The second digit in the code has a value exactly twice that of the first digit.**
3. **The value of the third digit in the code is less than the value of the fifth digit.**
4. **The fifth digit of an acceptable product code is 1.**

Given these constraints, let's determine which statements must be true.

### Step-by-Step Analysis

1. **Fifth Digit is 1:**
   Since the fifth digit is 1, we need to find the possible values for the remaining digits (0, 2, 3, 4).

2. **Second Digit Rule:**
   The second digit is twice the value of the first digit. Therefore, the possible pairs for the first and second digits are:
   - (0, 0) — not possible since each digit is unique.
   - (1, 2) — not possible because the fifth digit is 1.
   - (2, 4)
   - (3, 6) — not possible since 6 is not an allowed digit.
   - (4, 8) — not possible since 8 is not an allowed digit.

   So, the first and second digits must be (2, 4).

3. **Third and Fourth Digits:**
   We know the remaining digits to be used are 0, 3, and 4. The value of the third digit must be less than the value of the fifth digit (which is 1). Therefore, the possible digits for the third position are:
   - 0 (the only digit less than 1 from the remaining set)

4. **Remaining Digits:**
   The remaining digits to be placed are 3 and 4, and they must occupy the fourth and fifth positions, respectively.

   Now we have the digits as:
   - First: 2
   - Second: 4
   - Third: 0
   - Fourth: 3 (it must be 3 since 1 is already fixed at the fifth place)

### The code structure:
   The only possible configuration is 24031.

### Verification:
   - The first digit (2) is half of the second digit (4).
   - The third digit (0) is less than the fifth digit (1).
   - The digits used are 2, 4, 0, 3, 1, and each appears exactly once.

### Conclusion:
Given the fifth digit is 1, the following must be true:
1. **First digit is 2** (True, since the first digit must be 2 in the valid sequence 24031).
2. **Second digit is 0** (False, the second digit is 4).
3. **Third digit is 3** (False, the third digit is 0).
4. **Fourth digit is 4** (False, the fourth digit is 3).
5. **Fourth digit is 0** (False, the fourth digit is 3).

Therefore, the correct statement is:
1. (1) The first digit is 2.

Would you like any more details or have any questions? Here are five more questions you might find interesting:

1. How do you find the greatest common divisor of two numbers?
2. What are the properties of prime numbers?
3. How do you solve quadratic equations?
4. What is the difference between permutations and combinations?
5. How do you calculate the area of a circle?

**Tip:** Always break down complex problems into smaller, manageable steps and verify each part against the given conditions. This helps in identifying the correct solution systematically.

Explore the solution to a mathematical problem involving five-digit product codes. Understand the rules and constraints, including logical reasoning and basic arithmetic. Suitable for students in Grades 5-8.

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