Number Play - Numbers Tell us Things

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

$A = 7$ and $B = 6$

Looking at the units column: $A + 5$ results in a units digit of $2$. This means $A + 5 = 12$, which gives $A = 7$. Carrying over $1$ to the tens column: $1 + 3 + 2 = 6$. Therefore, $B = 6$.

If the 3-digit number $24x$ is divisible by $9$, what is the value of $x$?

$x = 3$

For a number to be divisible by $9$, the sum of its digits must be a multiple of $9$. Sum of digits $= 2 + 4 + x = 6 + x$. For $6 + x$ to be divisible by $9$, the smallest possible value for $x$ is $3$ (since $6 + 3 = 9$).

Show that the sum of the digits of a 2-digit number $ab$ and its reverse $ba$ is divisible by $11$. Use the number $47$.

$47 + 74 = 121$, and $121 = 11 \times 11$

Representing $47$ as $10(4) + 7$ and $74$ as $10(7) + 4$. Sum $= (40 + 7) + (70 + 4) = 110 + 11 = 121$. Since $121$ is $11 \times 11$, it is divisible by $11$.