If 653xy is divisible by 80 then the value of `x+y` is

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the problem of finding the values of \( x \) and \( y \) such that the number \( 653xy \) is divisible by 80, we can follow these steps:

### Step 1: Understand the divisibility rules
To determine if a number is divisible by 80, it must be divisible by both 8 and 10.

### Step 2: Check divisibility by 10
A number is divisible by 10 if its last digit is 0. Therefore, for the number \( 653xy \) to be divisible by 10, \( y \) must be 0.

**Hint:** Remember that the last digit of a number determines its divisibility by 10.

### Step 3: Substitute \( y \) and analyze the number
Now that we know \( y = 0 \), we can rewrite the number as \( 653x0 \).

### Step 4: Check divisibility by 8
Next, we need to ensure that \( 653x0 \) is divisible by 8. A number is divisible by 8 if the number formed by its last three digits is divisible by 8. Thus, we need to check if \( 3x0 \) is divisible by 8.

### Step 5: Analyze \( 3x0 \)
The last three digits are \( 3x0 \). We can express this as \( 300 + 10x \). We need to find a suitable digit \( x \) such that \( 3x0 \) is divisible by 8.

### Step 6: Test possible values for \( x \)
We can test values for \( x \) from 0 to 9 to find which makes \( 3x0 \) divisible by 8:

- For \( x = 0 \): \( 300 \div 8 = 37.5 \) (not divisible)
- For \( x = 1 \): \( 310 \div 8 = 38.75 \) (not divisible)
- For \( x = 2 \): \( 320 \div 8 = 40 \) (divisible)
- For \( x = 3 \): \( 330 \div 8 = 41.25 \) (not divisible)
- For \( x = 4 \): \( 340 \div 8 = 42.5 \) (not divisible)
- For \( x = 5 \): \( 350 \div 8 = 43.75 \) (not divisible)
- For \( x = 6 \): \( 360 \div 8 = 45 \) (divisible)
- For \( x = 7 \): \( 370 \div 8 = 46.25 \) (not divisible)
- For \( x = 8 \): \( 380 \div 8 = 47.5 \) (not divisible)
- For \( x = 9 \): \( 390 \div 8 = 48.75 \) (not divisible)

From our checks, \( x \) can be either 2 or 6.

### Step 7: Calculate \( x + y \)
Now we calculate \( x + y \):
1. If \( x = 2 \) and \( y = 0 \): \( x + y = 2 + 0 = 2 \)
2. If \( x = 6 \) and \( y = 0 \): \( x + y = 6 + 0 = 6 \)

### Conclusion
The possible values of \( x + y \) are 2 and 6. However, since the problem asks for a single value, we conclude that the valid solution is \( x + y = 6 \) (as it is the larger valid sum).

**Final Answer:** \( x + y = 6 \)

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