A Story of Numbers - The Idea of a Base

Find the values of $A$ and $B$ in the following addition: $\begin{array}{r} 3A \\ + 25 \\ \hline B2 \end{array}$

$A = 7$ and $B = 6$

In the units column, $A + 5$ results in a number ending in $2$. The only digit for $A$ that satisfies $A + 5 = 12$ is $A = 7$. We carry over $1$ to the tens column. In the tens column, $1 (carry) + 3 + 2 = B$, which gives $B = 6$.

Check if the difference between $74$ and its reverse is divisible by $9$.

$74 - 47 = 27$, which is $9 \times 3$.

Using the generalized form: $(10a + b) - (10b + a) = 9(a - b)$. Here $a=7$ and $b=4$. The difference is $9(7 - 4) = 9 \times 3 = 27$. Since $27$ is a multiple of $9$, it is divisible by $9$.

Write the number $582$ in expanded form using powers of $10$.

$5 \times 100 + 8 \times 10 + 2 \times 1$

In the base-10 system, the positions represent units ($10^0$), tens ($10^1$), and hundreds ($10^2$). Thus, $582 = (5 \times 10^2) + (8 \times 10^1) + (2 \times 10^0)$.