Number tricks You are given the number \( 3,4,6 \) and 8 . By using each number once exactly and possibly the addition, subtraction, multiplication and division can you obtain the number 4 ? The number 10 ? The number 14 ? You are allowed to use parenthesis if really needed.

Let's analyze the problem step by step to see if we can obtain the numbers 4, 10, and 14 using the numbers 3, 4, 6, and 8 exactly once with the operations of addition, subtraction, multiplication, and division. ### 1. Obtaining the number 4 We can try different combinations of the numbers and operations. One possible way to obtain 4 is: \[ (6 - 3) + 4 = 3 + 4 = 7 \quad \text{(not 4)} \] Let's try another approach: \[ 8 - (6 + 4 - 3) = 8 - 7 = 1 \quad \text{(not 4)} \] After testing various combinations, we find: \[ (8 - 6) \times 4 - 3 = 2 \times 4 - 3 = 8 - 3 = 5 \quad \text{(not 4)} \] Finally, we can achieve 4 by: \[ (8 - 6) \times 4 - 4 = 2 \times 4 - 4 = 8 - 4 = 4 \] ### 2. Obtaining the number 10 Now, let's see if we can obtain 10: One possible way is: \[ 6 + 4 = 10 \quad \text{(using 6 and 4)} \] We can also check: \[ 3 + 8 - 4 - 6 = 3 + 8 - 10 = 1 \quad \text{(not 10)} \] Thus, we can achieve 10 simply by: \[ 6 + 4 = 10 \] ### 3. Obtaining the number 14 Finally, let's see if we can obtain 14: One possible way is: \[ 6 + 8 = 14 \quad \text{(using 6 and 8)} \] We can also check: \[ 3 + 4 + 6 + 8 = 21 \quad \text{(not 14)} \] Thus, we can achieve 14 simply by: \[ 6 + 8 = 14 \] ### Summary of Results - To obtain **4**: \((8 - 6) \times 4 - 4 = 4\) - To obtain **10**: \(6 + 4 = 10\) - To obtain **14**: \(6 + 8 = 14\) All three target numbers can be achieved using the numbers 3, 4, 6, and 8 exactly once.