Understanding Divisibility by Multiple Numbers

A number is divisible by two or more numbers if it is divisible by their Least Common Multiple (LCM). In this case, we need to find a number that is divisible by both 27 and 19.

Calculating the LCM of 27 and 19

First, we find the LCM of 27 and 19. Since 19 is a prime number and 27 ($27 = 3^3$) does not have 19 as a factor, they are relatively prime (their Greatest Common Divisor is 1).

Therefore, the LCM of 27 and 19 is simply their product:

$$ \text{LCM}(27, 19) = 27 \times 19 $$

Let's calculate the product:

$$ 27 \times 19 = 27 \times (20 - 1) = (27 \times 20) - (27 \times 1) = 540 - 27 = 513 $$

So, any number divisible by both 27 and 19 must be divisible by 513.

Checking the Options for Divisibility by 513

Now, we will check each option to see if it is divisible by 513.

Conclusion

From the calculations above, only the number 34371 is perfectly divisible by 513. This means 34371 is divisible by both 27 and 19.

Alternatively, we could check divisibility by 27 and 19 separately:

• 34371: Sum of digits = $3+4+3+7+1 = 18$. Since 18 is divisible by 9, 34371 is divisible by 9. For divisibility by 27, we check $34371 \div 27 = 1273$. It is divisible by 27. For divisibility by 19, we check $34371 \div 19 = 1809$. It is divisible by 19.

Since 34371 meets both conditions, it is the correct answer.