+0

# Base Numbers

0
202
1

A base-10 integer n can be represented \$32_a\$ as in one base and \$13_b\$ in another base, where a and b are any integer bases larger than 3. What is the smallest possible sum a+b?

Jun 12, 2021

### 1+0 Answers

#1
+288
+3

By assumption \$a \ge4\$ and \$b\ge4\$, and the assumption \$32_a = 13_b\$ gives the constraint \$3a+2=b+3\$, that is, \$a = (b+1)/3\$.   Solving \$a=(b+1)/3 \ge 4\$ gives \$b\ge11\$.  When \$b=11\$ we have \$a=(11+1)/3=4\$.    So the smallest possible sum \$a+b=4+11=15\$.

Jun 12, 2021