Is 34 divisible by both 3 and 9?
To determine if \(34\) is divisible by \(3\) and \(9\), we need to apply the standard rules of divisibility. A number is divisible by another if the division results in a whole number with no remainder. Alternatively, we can use the sum of digits rule: a number is divisible by \(3\) or \(9\) if the sum of its digits is divisible by \(3\) or \(9\), respectively. The number given is \(34\). The sum of its digits is \(3 + 4 = 7\).
To check if \(34\) is divisible by \(3\), we divide \(34\) by \(3\). Dividing \(34\) by \(3\) yields \(11\) with a remainder of \(1\), since \(3 \times 11 = 33\) and \(34 - 33 = 1\). Similarly, applying the sum of digits rule, we know that the sum of the digits of \(34\) is \(7\). Since \(7\) is not divisible by \(3\), the number \(34\) itself cannot be divisible by \(3\).
To check if \(34\) is divisible by \(9\), we divide \(34\) by \(9\). Dividing \(34\) by \(9\) gives \(3\) with a remainder of \(7\), since \(9 \times 3 = 27\) and \(34 - 27 = 7\). Using the sum of digits rule, since the sum of the digits of \(34\) is \(7\), and \(7\) is not divisible by \(9\), the number \(34\) is not divisible by \(9\).
Based on the calculations above, \(34\) is not divisible by \(3\), and it is not divisible by \(9\). Therefore, it is not divisible by either. The answer is \(\boxed{\text{False}}\).
Divisibility rules are extremely useful in daily life for quick mental math, such as checking if items can be evenly distributed among groups. For instance, if you have 34 cookies and want to split them equally among 3 friends, the fact that 34 is not divisible by 3 tells you immediately that there will be leftovers. Knowing these rules allows you to manage resources efficiently without needing a calculator!