The average of five distinct integers is 60. If the largest integer is 70, what is the minimum possible value of the smallest integer ?

Options

A. 0
B. 1
C. 20
D. 26

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Understand the problem
We are given that the average of five distinct integers is 60, and the largest integer among them is 70. We need to find the minimum possible value of the smallest integer.

### Step 2: Set up the equation for the average
The average of five integers \( a, b, c, d, e \) can be expressed as:
\[ \frac{a + b + c + d + e}{5} = 60 \]
Multiplying both sides by 5 gives:
\[ a + b + c + d + e = 300 \]

### Step 3: Substitute the largest integer
We know that the largest integer \( a \) is 70. Therefore, we can substitute \( a \) in the equation:
\[ 70 + b + c + d + e = 300 \]
This simplifies to:
\[ b + c + d + e = 300 - 70 = 230 \]

### Step 4: Minimize the smallest integer
Let \( e \) be the smallest integer. To minimize \( e \), we need to maximize \( b, c, \) and \( d \) while ensuring they are distinct and less than 70.

### Step 5: Choose the largest possible distinct integers
The largest integers less than 70 that are distinct are:
- \( b = 69 \)
- \( c = 68 \)
- \( d = 67 \)

### Step 6: Substitute these values into the equation
Now we substitute \( b, c, \) and \( d \) into the equation:
\[ 69 + 68 + 67 + e = 230 \]
Calculating the sum:
\[ 204 + e = 230 \]
This leads to:
\[ e = 230 - 204 = 26 \]

### Conclusion
Thus, the minimum possible value of the smallest integer \( e \) is:
\[ \boxed{26} \]