1. Factors and Multiples #5 next 3 ×6 18 Taking the Fear out of Math
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Factors of a Number
There is a saying to the effect that it is easier to scramble an egg than to
unscramble it.
This saying can be applied to the way most people tend to learn their multiplication facts. They tend to memorize the facts in the order in which they are read, that is, from left to right.
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For example, when they see 5 × 4 = 20, they remember that when you multiply 5 by 4 the answer is 20. However, it is not as obvious to them that they also see that 20 is the product of 4 and 5.
The problem is further compounded by the fact that while 5 × 4 = 20, there are other pairs of whole numbers whose product is 20 (namely, 20 and 1, and 10 and 2).
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4. Definition
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When we start with 20 and rewrite it, for example, as 5×4, we refer to the process as factoring 20, and we refer to 4 and 5 as being factors (or divisors) of 20.
The factors of 20 are 1, 2, 4, 5, 10, and 20.
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5. A Visual Way to Find the Factors of 20
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A Visual Way to Find the Factors of 20
Given 20 square tiles one can be asked to find all the rectangular arrays that can be formed by the tiles.
For example, with 20 tiles you should discover that the only rectangular arrays that can be made are a 1 by 20, 20 by 1,
2 by 10, 10 by 2, 4 by 5, and 5 by 4.
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6. Given 20 square tiles, we see that…
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A Visual Way to Find the Factors of 20
Given 20 square tiles, we see that…
20 × 1
5 × 4
10 × 2
1 × 20
2 × 10
4 × 5
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7. Students often learn that 5×4 = 20 by “skip counting”.
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Multiples of a Number
Students often learn that 5×4 = 20 by “skip counting”.
More specifically, they internalize that 5×4 means the sum of 5 four’s and they then count by 4’s to obtain…
4, 8, 12, 16, 20
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8. More formally, they have listed the first
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More formally, they have listed the first
5 multiples of 4…
1×4, 2×4, 3×4, 4×4, and 5×4.
The fact that 5×4 = 4×5 means that 20 is also the fourth multiple of 5.
5, 10, 15, 20
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9. 15th multiple of 4 and the 12th multiple of 5.
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So we refer to 20 as a common multiple of 4 and 5. What this means is that if we list the multiples of 4 and the multiples of 5, 20 will appear on both lists. In fact, since 20 is the least number that appears on both lists we call 20 the least common multiple of 4 and 5.1
note
1 Notice that once we know that 20 is a common multiple of 4 and 5 we also know that any multiple of 20 is also a common multiple of 4 and 5.
For example, 3 × 20 = 3 × 4 × 5 = 4 × (3 × 5) = 5 × (4 × 3). In other words we can write 60 as 4 × 15 and as 5 ×12, which means that 60 is the
15th multiple of 4 and the 12th multiple of 5.
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A Generalization
The product of two whole numbers is always a common multiple of both numbers.
For example, without knowing the numerical value of 217×349 we know for sure that it’s the 217th multiple of 349 and the 349th multiple of 217.
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Note
Too often students compute such products as 217×349 by rote memory without internalizing what it means in terms of common multiples. This tends to hamper their ability to use mathematics to solve “real world” problems.
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An Application
Hot dogs usually come in packages of 10 while hot dog buns usually come
in packages of 8.
Suppose we want to order the correct number of packages of each so that we have the same number of hot dogs and buns.
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Since hot dogs come in packages of 10, the number of hot dogs will always be a multiple of 10.
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120,
Since the buns come in packages of 8, the number of buns will always be a multiple of 8.
8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112,
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14. 5 packages of buns, we will have 40 of each,
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120,
Hot Dogs
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8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112,
Buns
From the two lists of multiples we see that if we buy 4 packages of hot dogs and
5 packages of buns, we will have 40 of
each,
if we buy 8 packages of hot dogs and 10 packages of buns, we will have 80 of each,
and if we buy 12 packages of hot dogs and 15 packages of buns, we will have 120 of each.
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Note
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120,
Hot Dogs
8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112,
Buns
Notice that 40 is the least number that appears on each of the two lists. Hence, we refer to it as the least common multiple of 8 and 10, and therefore, any multiple of 40 is also a multiple of both 8 and 10.
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16. Indeed, as we can see from our two lists,
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120,
Hot Dogs
8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112,
Buns
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Indeed, as we can see from our two lists,
80 is the 10th multiple of 8 and the 8th multiple of 10. However, it is not the least common multiple of 8 and 10. It is one example of the fact that since 40 is the least common multiple of 8 and 10, every multiple of 40 is also a common multiple of 8 and 10.
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While it is more convenient from a computational point of view to deal with the least common multiple of a group of numbers, it is sometimes quite cumbersome to find it. 
For example, we can find a common multiple of 40, 45, and 72 by multiplying the three numbers to obtain 87,600 as being one such number.
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And to find the least common multiple of 40, 45, and 72, we could list the multiples of each and see what is the first number that appears on each of the three lists. 
We would find…
40, 80, 120, 160, 200, 240, 280, 320, 360, 400, 420…
45, 90, 135, 180, 225, 270, 315, 360, 405, 450…
72, 144, 216, 288, 360, 432, 504…
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Clearly it is easier to do various computations using 360 than it would be using 87,600.  However, listing the multiples can be quite cumbersome, especially when we deal with much larger groups of numbers. 
A convenient way to eliminate much of the tedium that accompanies listing the multiples is by introducing
prime numbers.
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Factors and Multiples
In the next presentation we will talk about prime numbers and prime factorization.
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