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8 and n as factors, which expression has both of these?

This question aims to find an expression that has both the given factors. Moreover, it is helpful to have a number divisible by the given numbers.

This question is based on the concepts of arithmetic, and the factors of a number include all divisors of that specific number. The factors of the number 16, for example, are 1, 2, 4, and 16. We can obtain another whole integer number by dividing 16 with any of the numbers given above.

Expert Answer

We are looking for an expression that has 8 and $ n $ as factors. Therefore, suppose that $ E $ is the expression that has a factor, which means that the expression is divisible by 8.
Hence,
\[ E (X) = 8 X .  ( n )^X \]
Where $ X $ is any positive integer $ n $.
\[ E (X) = 8 X  ( n )^X \]

Alternate Solution

From the question, we have $ 8 $ and $ n $ as factors of an expression. Moreover, these factors should be present in the expression. The example is as follows:

\[ x = 8 + n \]

Numerical Results

The expression that has both 8 and n as factors is as follows.

\[ E (X) = 8 X  ( n )^X \]

or an alternate solution could be:

\[ x = 8 + n \]

Example

We have a number 8 with exactly four different factors, including 1, 2, 4, and 8. Therefore, if you have a number 36, how many factors does it have?

Solution

The number 8 has 1, 2, 4, and 8; exactly four factors. Therefore, we can find different factors of 36 as shown below.
Step 1: Total number of factors number 36 can be calculated as follows:

\[ 36 = 2 \times 2 \times 3 \times 3 \]

\[ 36 = 2^2 \times 3^2 \]

\[ (36) = ( 2 + 1 ) \times ( 2 + 1 )\]

\[ = 3 \times 3 \]

\[ = 9 \]

So the number 36 has exactly 9 factors.

Step 2: The number of factors of the number 36 are as follows:

$ 1 \times 36 = 36 $

$ 2 \times 18 = 36 $

$ 3 \times 12 = 36 $

$ 4 \times 9 = 36 $

$ 6 \times 6 = 36  $

$ 9 \times 4 = 36 $

$ 12 \times 3 = 36 $

$ 18 \times 2 = 36 $

$ 36 \times 1 = 36 $

With this, the factors of 36 are  1, 2, 3, 4, 6, 9, 12, 18, and 36.

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