To find the relationship between the numbers in the equations \(4 \# 2 = 16\) and \(1 \# 5 = 13\), we need to look for a pattern or operation that consistently transforms the pairs of numbers on the left into the result on the right.

Let's analyze the given equations:

1. \(4 \# 2 = 16\)
2. \(1 \# 5 = 13\)

Let's hypothesize some operations and test them:

### Hypothesis 1: Addition and Multiplication
- \(4 \# 2\)
  - \(4 + 2 = 6\)
  - \(4 \times 2 = 8\)
  - Neither addition nor multiplication alone gives 16.

### Hypothesis 2: Powers
- \(4^2 = 16\), but we need to involve both numbers.
- Maybe: \(a^2 + b\)
  - \(4^2 + 2 = 16 + 2 = 18\), doesn't match.
  - \(a^2 - b\)
    - \(4^2 - 2 = 16 - 2 = 14\), doesn't match.

### Hypothesis 3: Combination
Maybe the operation is a combination of addition, multiplication, or some function applied to one of the numbers and then combined.

Let’s try:

### Hypothesis 4: Linear Combination
Check if \(a \# b = ka + lb\):

- For \(4 \# 2 = 16\)
  - \(4k + 2l = 16\)

- For \(1 \# 5 = 13\)
  - \(1k + 5l = 13\)

This can be solved as a system of linear equations:

1. \(4k + 2l = 16\)
2. \(1k + 5l = 13\)

Let's solve this system of equations:
\[
\begin{cases}
4k + 2l = 16 \quad \text{(i)}\\
k + 5l = 13 \quad \text{(ii)}
\end{cases}
\]

Multiply equation (ii) by 4:
\[
4k + 20l = 52 \quad \text{(iii)}
\]

Subtract equation (i) from (iii):
\[
4k + 20l - (4k + 2l) = 52 - 16 \\
18l = 36 \\
l = 2
\]

Substitute \(l = 2\) back into equation (ii):
\[
k + 5(2) = 13 \\
k + 10 = 13 \\
k = 3
\]

So, the relationship appears to be:
\[
a \# b = 3a + 2b
\]

Let's verify:
- \(4 \# 2 = 3(4) + 2(2) = 12 + 4 = 16\)
- \(1 \# 5 = 3(1) + 2(5) = 3 + 10 = 13\)

This relationship holds true. 

### Conclusion
The relationship for the operation \(\#\) is \(a \# b = 3a + 2b\).

Would you like any further details or have any questions? 

Here are some related questions to expand your understanding:

1. What is \(3 \# 4\)?
2. If \(a \# b = 20\), and \(a = 2\), what is \(b\)?
3. How can you describe the operation \(\#\) in words?
4. What is the result of \(0 \# 0\)?
5. Is the operation \(\#\) commutative?
6. What happens if one of the numbers in the operation \(\#\) is negative?
7. What is \(5 \# 3\)?
8. If \(a \# b = 3a + 2b\), solve for \(b\) in terms of \(a\) and \(\#\).

**Tip:** When trying to find a pattern or relationship, break down the problem into smaller parts and test different hypotheses until you find a consistent rule.

Explore and understand the relationship between the numbers in equations 4#2=16 and 1#5=13. Learn how to analyze and solve such problems involving arithmetic operations and linear combinations. Suitable for high school students and those interested in mathematical problem-solving.

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