Question

i have 4 numbers -- a, b, c, and $d--$ that i can pair up in 6 different ways to multiply -- $a^* b, a^* c$, $a^* d, b^* c, b^* d$, and $c^* d$. when i do, i get products in the set $\{5,8,10,16,32, x\}$. what does $x$ equal? please input your answer as a decimal rounded to the nearest hundreth.

Answer 1

By factorizing the given products and deducing what the original four numbers must be, the missing product x is found to be the multiplication of 5 and 32, which equals 160.00 when rounded to the nearest hundredth.

Step-by-step explanation:

To find the unknown product x, we can analyze the given products which are {5, 8, 10, 16, 32, x}. Notice that all numbers except x are products of prime factors 2 and/or 5. Since the same four numbers are used to produce these products, x must also be a product of the primes 2 and 5. Start by factorizing each number:

• 5 is 5
• 8 is 2^3
• 10 is 2 × 5
• 16 is 2^4
• 32 is 2^5

We can pair the original numbers a, b, c, and d in such a way to get the products above. For instance, since 32 is 2^5, one of the numbers (say a) must be 32, and the numbers we are multiplying by must include 1 to get the other products. Thus, one of the other numbers (say d) must be 1. To get 16, we then know that another number must be 2, and the last number must be 5 to get the product of 10 when multiplied by 2. They can be combined in pairs to get all the other numbers.

Therefore, the numbers are 1, 2, 5, and 32, and the missing product x is 5 multiplied by 32, which is 160. So, x = 160.00, rounded to the nearest hundredth.