Question

Find the sum of all the possible values of \( p \) such that \( p^4 - p^3 \) has the unit’s digit as 2, where \( 20 \leq p \leq 30 \).

• A. 46
• B. 49
• C. 53
• D. 56
• E. 50

Hint:

When solving such problems, focus on checking the unit's digit of the expression to match the given condition.

Answer 1

The Correct Option is C

Step 1: Identifying the condition.
We are given the condition that the unit's digit of \( p^4 - p^3 \) is 2, where \( 20 \leq p \leq 30 \). 
Step 2: Checking each value of \( p \). 
We need to calculate \( p^4 - p^3 \) for values of \( p \) from 20 to 30 and check for the unit's digit being 2. After calculating for all values, the following results hold: - For \( p = 24 \), \( p^4 - p^3 \) has unit's digit 2. - For \( p = 29 \), \( p^4 - p^3 \) has unit's digit 2. Thus, the possible values of \( p \) are 24 and 29. 
Step 3: Summing the possible values of \( p \). 
The sum of the possible values of \( p \) is: \[ 24 + 29 = 53 \] Final Answer: \[ \boxed{53} \]