Question

Which is a complete list of factors of each term in the expression 15 + 20 x?

Answer 1

Answer:

  A:

     15: 1, 3, 5, 15

     20x: 1, 2, 4, 5, 10, 20, x

Step-by-step explanation:

When finding divisors of a number, it is often convenient to start with the prime factors. Those may be raised to some power.

For the numbers here, the prime factors are ...

  15 = 3·5

  20 = 2²·5

One of the reasons this is useful is that each of these prime factors has an exponent. The number of divisors is the product of these exponents incremented by 1. That is, the number of divisors will be ...

  for 15: (1+1)(1+1) = 4

  for 20: (2+1)(1+1) = 6

Knowing this can help you make sure you have found all of the divisors. This number includes 1 and the number itself.

Then the divisors are ...

  15 ⇒ 1, 3, 5, 15 . . . . . . . . . . . 4 divisors

  20 ⇒ 1, 2, 4, 5, 10, 20 . . . . . 6 divisors

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For the variables, it is a little hard to tell what is intended. At least, a list of the individual variables is required. Hence x is added to the list for 20x.

The "complete" list of factors is ...

• 15: 1, 3, 5, 15
• 20x: 1, 2, 4, 5, 10, 20, x

_____

Further comment on variable factors

If the variables were to be included in the divisor list the way the prime factors are, then we would first start by listing the prime factors:

 {2, 2, 5, x}

Then we would form the possible subsets:

  { }, {2}, {5}, {x}, {2, 2}, {2, 5}, {2, x}, {5, x}, {2, 2, 5}, {2, 2, x}, {2, 5, x}, {2, 2, 5, x}

and finally, form the products of the contents of these subsets:

  1, 2, 5, x, 4, 10, 2x, 5x, 10, 4x, 10x, 20x

Apparently, the answer choices for this question do not recognize 2x as a factor of the second term. The treatment for numbers is obviously different from the treatment for variables, but it is hard to tell what the rules for variables are from the answer choices here. (Consult your teacher and/or text.)