Find the units' digit in the expression `(515)^(31)+(525)^(90)?`

Answer

To find the unit's digit of the expression \( (515)^{31} + (525)^{90} \), we can follow these steps:

### Step 1: Identify the unit's digits of the base numbers
The unit's digit of \( 515 \) is \( 5 \) and the unit's digit of \( 525 \) is also \( 5 \).

### Step 2: Determine the exponent for each term
- For \( (515)^{31} \), the exponent is \( 31 \).
- For \( (525)^{90} \), the exponent is \( 90 \).

### Step 3: Find the unit's digit pattern for powers of \( 5 \)
The unit's digit of any power of \( 5 \) is always \( 5 \):
- \( 5^1 = 5 \)
- \( 5^2 = 25 \) (unit's digit is \( 5 \))
- \( 5^3 = 125 \) (unit's digit is \( 5 \))
- \( 5^4 = 625 \) (unit's digit is \( 5 \))
- And so on...

Thus, both \( (515)^{31} \) and \( (525)^{90} \) will have a unit's digit of \( 5 \).

### Step 4: Add the unit's digits
Now, we add the unit's digits:
- Unit's digit of \( (515)^{31} \) is \( 5 \)
- Unit's digit of \( (525)^{90} \) is \( 5 \)

So, \( 5 + 5 = 10 \).

### Step 5: Determine the unit's digit of the sum
The unit's digit of \( 10 \) is \( 0 \).

### Final Answer
Therefore, the unit's digit in the expression \( (515)^{31} + (525)^{90} \) is \( 0 \).

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