Download A Method for Finding the Number of Factors for Numbers and Some

Finding the Number of Factors for a Given Number
For many problems, identifying the factors of a number is a necessary skill, especially in middle
school. It’s an important skill when working with fractions, and it comes in handy in algebra,
too. For small numbers, listing the factors is not very difficult. The number 12, for instance, has
6 factors: 1, 2, 3, 4, 6 and 12. However, as numbers increase and have more factors, listing them
becomes more difficult. By knowing ahead of time how many factors a number has, listing them
is made easier; or, at least, we’ll know if we’ve found all of them.
The prime factorization of 12 is 22 ⋅ 31. This tells us that when identifying the factors of 12, we
need to find every combination of 2p, such that 0 < p < 2, and 3q, such that 0 < q < 1. We can find
every combination of these prime factors by arranging them in a chart as seen below. This chart
will contain all of the factors of 12:
20
21
22
30
1
2
4
31
3
6
12
If the distinct prime factors have
exponents of p and q, then there are p + 1
columns and q + 1 rows in our chart and a
total of (p + 1)(q + 1) factors.
A residual result of this fact is that all square numbers, and only square numbers, have an odd
number of factors. If a number is a perfect square, then each of its prime factors will have an
even exponent. Adding 1 to each exponent will give an odd number, and when odd numbers are
multiplied, the result is odd. For instance, the perfect square 144 has the prime factorization
24 · 32. The number of factors, then, is (4 + 1)(2 + 1) = 15. More generally, a perfect square will
have a prime factorization of the form p2a · q2b. The number of factors is (2a + 1)(2b + 1), which
will always have an odd product.
What if the number has more than two prime factors?
Now, consider a number with three distinct prime factors. The prime factorization of 90
is 21 ⋅ 32 ⋅ 51. Therefore, just using the prime factors 2 and 3, we know that there are at least
6 factors. If each of those 6 factors is combined with 50 and 51, we’ll have 6 ⋅ 2 = 12 factors
for 90. Notice, this is the same as seeing that the exponents of the distinct prime factors are p, q,
and r, and therefore, there are (p + 1)(q + 1)(r + 1) total factors.
It may also help to visualize the factors of 90 with the picture below—there are two possible
exponents for 2, two possible exponents for 5, and three possible exponents for 3, giving a
2 × 2 × 3 cube, which consists of 12 smaller cubes. Each of the smaller cubes represents a factor
of 90. The highlighted cube, for instance, represents the factor 20 × 31 × 51 = 15.
32
31
21
30
5
0
5
1
20
Using the ideas above, there is an algorithm for determining the number of factors for any
positive integer.
How many factors does 1260 have?
1. 1260 = 22 ⋅ 32 ⋅ 51 ⋅ 71
2.
add 1 to each exponent: 3, 3, 2, 2
3. 3 ⋅ 3 ⋅ 2 ⋅ 2 = 36 factors for 1260
To find the number of factors for a particular
number:
1. identify the prime factorization of the
number
2. add 1 to each of the exponents of the
distinct prime factors of the number
3. find the product of these new
“exponents”
This algorithm will be helpful in solving many MATHCOUNTS problems which simply ask for
the number of factors. It will also be useful, however, in checking your work—for instance, if we
wanted to know the sum of the 18th and 19th factors of 1260 when they are listed in order from
least to greatest, knowing that there are 36 factors would help tremendously.
Sample MATHCOUNTS Problems Involving Factors
1. How many factors does the expression 4 x5 y3 have, if x and y are distinct, odd primes?
2. How many factors do 840 and 12,250 have in common?