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

Step 1: $p^4-p^3=p^3(p-1)$; its units digit depends only on the units digit $d$ of $p$ (with the units digit of $p-1$ being $d-1$, or $9$ if $d=0$).
Step 2: Build a table of the units digit of $p^3(p-1)$ for each possible last digit $d$ of $p$: $d=0\to0$; $d=1\to0$; $d=2\to8\times1=8$; $d=3\to7\times2=14\to4$; $d=4\to4\times3=12\to2$; $d=5\to5\times4=20\to0$; $d=6\to6\times5=30\to0$; $d=7\to3\times6=18\to8$; $d=8\to2\times7=14\to4$; $d=9\to9\times8=72\to2$.
Step 3: Only last digits $4$ and $9$ give a units digit of $2$.
Step 4: In the range $20$ to $30$, the numbers ending in $4$ or $9$ are $24$ and $29$.
Step 5: Sum $=24+29=53$.
\[\boxed{53}\]