Question

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

• A. 46
• B. 49
• C. 53
• D. 56

Hint:

The last digit of $p^4-p^3$ depends only on the last digit of $p$; build a small table of last-digit outcomes for $p^3(p-1)$.

Answer 1

The Correct Option is C

\(p^4-p^3=p^3(p-1)\); the unit digit depends only on the unit digit of \(p\). Testing 20 to 30: for \(p=24\), unit of \(4^3=4\) times unit of \(23=3\) gives \(12\to 2\). For \(p=29\), unit of \(9^3=9\) times unit of \(28=8\) gives \(72\to 2\). No other value in the range works. Sum \(=24+29=53\). Correct option: 53.