18. Factors · Mastering AMC 8

18.10%

Number of Factors

In this section you will be able to count the factors of a number straight from its prime factorization.

How many numbers divide 112 evenly? You could test 1, 2, 3, 4 and on and on. Or you could draw a tiny grid and count the dots!

Every factor of $2^4 \times 7$ is made by choosing a power of 2 and a power of 7. You have 5 choices for the first and 2 for the second. So there are $5 \times 2 = 10$ factors.

Look:each exponent can be anything from 0 up to its value. That’s why you add 1 before you multiply.

Exploreplay first, formulas later

Try this: This grid shows 72 = 2³ × 3². Click a few dots. Is every dot a factor of 72? How many columns and rows are there?
Try this: Raise the exponent of 2 by one. How many dots appear? Why does the count go up by a whole row's worth?
Try this: Press + Prime. The grid becomes a stack of grids. How many grids are there, and how many dots in all?
Try this: Press Squares. Which dots stay lit? Look at their exponents.

Name it

Theorem 18.1.1 · Number of Factors of a Number

If the prime factorization of the number is expressed as:

p_1^{e_1} \times p_2^{e_2} \times \cdots \times p_k^{e_k}

then the number of factors of this number is

(e_1 + 1)(e_2 + 1) \cdots (e_k + 1)

Concept 18.1.2

Basically, in order to find the number of factors of a number:

Find the prime factorization of the number
Add 1 to all of the exponents
Multiply them together

Concept 18.1.3

Here is a logical way to think about this formula: for each prime $p_k$ , the exponent can be anything from $0$ to $e_k$ . Therefore, there are $e_k + 1$ choices. Therefore, the total number of factors is just the product of all such $e_k + 1$ terms.

For example, in $112 = 2^4 \times 7$ , in the factor, the power of $2$ can be $2^0$ , $2^1$ , $2^2$ , $2^3$ , or $2^4$ . The power of $7$ can be $7^0$ or $7^1$ .

Therefore, in total there would be $5 \times 2 = 10$ ways just as we calculated in the previous problem. This logic will be useful when other conditions are imposed.

Watch itpredict, then reveal

Example 18.1

How many factors does the number 112 have?

1
What is the prime factorization of 112?

Predict first:

Example 18.2

How many positive factors does the number 144 have that are perfect squares?

1
What is the prime factorization of 144?

Example 18.3

AMC 8

How many positive integer factors of 2020 have more than 3 factors? (As an example, 12 has 6 factors, namely 1, 2, 3, 4, 6, and 12.)

1
What is the prime factorization of 2020?

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Try it

Problem 18.4.8(additional)

How many positive factors does the number 60 have?

Problem 18.4.1

AMC 8

What is the sum of the prime factors of 2010?

Contest cornerreal contest problems

Problem 18.4.2

AMC 8

How many positive factors does 23,232 have?

Problem 18.4.10(additional)

EMCC

How many positive divisors of 2020 do not also divide 1010?

Problem 18.4.3

AMC 8

On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process continues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in the group?

Problem 18.4.4

Omega Learn Math Contest

How many positive multiples of 1,001 have less than 15 factors?

Problem 18.4.12(additional)

MATHCOUNTS

A positive integer $q$ is the product of a prime number and a perfect square. Additionally, $q$ is the product of a different prime number and a perfect cube. What is the least possible value of $q$ ?

Checkpoint for section 18.1

Get 2 of 3 right to master this section. One try each, no hints.

18.20%

Sum of Factors

In this section you will be able to add up all the factors of a number without listing them.

Add up every factor of 112. Listing all ten and adding them works. But what if the number had 200 factors? There’s a shortcut that uses just two small sums.

Add up the powers of each prime. Then multiply those sums together. When you expand the product, every factor appears exactly once.

Exploreplay first, formulas later

Try this: Read the sums of powers under the grid. Multiply them. Does that match the total?
Try this: Expand (1 + 2 + 4 + 8)(1 + 3 + 9) in your head. Why does each dot show up exactly once?
Try this: Press Odd, then Even. Which sum of powers changes? Which stays the same?

Name it

Theorem 18.2.1 · Sum of Factors of a Number

If the prime factorization of the number is expressed as:

p_1^{e_1} \times p_2^{e_2} \times \cdots \times p_k^{e_k}

then the sum of factors of this number is

(1 + p_1^1 + p_1^2 + \cdots + p_1^{e_1 - 1} + p_1^{e_1})(1 + p_2^1 + p_2^2 + \cdots + p_2^{e_2 - 1} + p_2^{e_2}) \cdots (1 + p_k^1 + p_k^2 + \cdots + p_k^{e_k - 1} + p_k^{e_k})

Concept 18.2.2 · Sum of Factors of a Number

Essentially, for a prime $p$ in the prime factorization, first find the sum of $p^k$ for all possible exponents $k$ in the prime factorization. Then, we will multiply all such sums for all of the primes to get the sum of all the factors of the number. This may seem complicated, but it will make more sense with an example.

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Example 18.4

What is the sum of factors of 112?

1
What is the prime factorization of 112?

Try it

Problem 18.4.9(additional)

Find the sum of factors of 320.

Contest cornerreal contest problems

Problem 18.4.5

AMC 10

Let $N = 34 \cdot 34 \cdot 63 \cdot 270$ . What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$ ?

Problem 18.4.11(additional)

BmMT

What is the sum of $\frac{1}{a}$ over all positive factors $a$ of the number 360?

Checkpoint for section 18.2

Get 2 of 3 right to master this section. One try each, no hints.

18.30%

Product of Factors

In this section you will be able to multiply all the factors of a number by pairing them up.

Multiply all the factors of 20: 1 × 2 × 4 × 5 × 10 × 20. That’s a big number. Can you find it without multiplying six things?

Factors come in pairs: $1 \times 20$ , $2 \times 10$ , $4 \times 5$ . Each pair multiplies to the number itself. With $f$ factors you get $f/2$ pairs.

Look:on the grid, partners sit opposite each other across the centre.

Exploreplay first, formulas later

Try this: The dot for 1 is picked. Where is its partner? What do the two multiply to?
Try this: Click other dots. Does every pair multiply to 20? Where is the centre of the grid?
Try this: Make the number a perfect square, like 2² × 3². Which dot is its own partner?

Name it

Theorem 18.3.1 · Product of Factors of a Number

The product of factors of a number $n$ where it has $f$ factors (this can be calculated using the number of factors formula) is

n^{\frac{f}{2}}

Watch itpredict, then reveal

Example 18.5

Find the product of the factors of 20.

1
What is the prime factorization of 20?

Example 18.6

Omega Learn

Let $a$ be the product of all odd factors of 54. Let $b$ be the sum of all even factors of 36. Find $a - b$ .

1
$54 = 2 \times 3^3$ . How many odd factors does 54 have?

Predict first:
Your prediction

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Contest cornerreal contest problems

Problem 18.4.6

AIME

Let $n = 2^{31} 3^{19}$ . How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$ ?

Checkpoint for section 18.3

Get 2 of 3 right to master this section. One try each, no hints.

18.4

Practice Problems

In this section you will be able to mix counting, summing and exponent tricks on contest problems.

Two tougher ones to finish! Start every problem with a prime factorization. Every problem has three hints if you get stuck.

Want every problem from this chapter on one page? Open the problem set.

Contest cornerreal contest problems

Problem 18.4.7

AMC 10

For some positive integer $n$ , the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$ . How many positive integer divisors does the number $81n^4$ have?

Problem 18.4.13(additional)

MATHCOUNTS

What is the greatest integer $k$ such that $80!$ is divisible by $45^k$ ?