8 points] If $$ A $$ and $$ B $$ are digits such that $$ \overline{A 149 B} $$ is divisible by 72 , find the value of the product $$ A \cdot B $$.

Question

UpStudy AI · Accepted Answer

To determine the product $$ A \cdot B $$ such that the number $$ \overline{A149B} $$ is divisible by 72, we need to ensure that the number satisfies the divisibility rules for both 8 and 9, since $$ 72 = 8 \times 9 $$ and 8 and 9 are coprime.

### Step 1: Divisibility by 8
A number is divisible by 8 if its last three digits form a number that is divisible by 8. For $$ \overline{A149B} $$, we look at the last three digits, which are $$ 49B $$.

We need to find $$ B $$ such that $$ 49B $$ is divisible by 8:
$$
49B = 490 + B
$$
To find $$ B $$ that satisfies $$ 490 + B \equiv 0 \pmod{8} $$:
$$
490 \div 8 = 61 \text{ with a remainder of } 2 \implies 490 \equiv 2 \pmod{8}
$$
Thus,
$$
2 + B \equiv 0 \pmod{8} \implies B \equiv 6 \pmod{8}
$$
Since $$ B $$ is a single digit, $$ B = 6 $$.

### Step 2: Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits of $$ \overline{A1496} $$ is:
$$
A + 1 + 4 + 9 + 6 = A + 20
$$
We need $$ A + 20 $$ to be divisible by 9:
$$
A + 20 \equiv 0 \pmod{9} \implies A + 2 \equiv 0 \pmod{9} \implies A \equiv 7 \pmod{9}
$$
Since $$ A $$ is a single digit, $$ A = 7 $$.

### Conclusion
With $$ A = 7 $$ and $$ B = 6 $$, the product is:
$$
A \cdot B = 7 \times 6 = 42
$$

**Answer:** 42
$$ A = 7 $$ and $$ B = 6 $$, so $$ A \cdot B = 42 $$.

8 points] If \( A \) and \( B \) are digits such that \( \overline{A 149 B} \) is divisible by 72 , find the value of the product \( A \cdot B \).

Solución de tutoría real

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Arithmetic Questions