David W. answered • 10/05/15
Tutor
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Experienced Prof
Prime factors are very important in math; they are useful for simple tasks like finding the LCD (least common denominator) and for very complicated tasks like decrypting coded messages between computers.
The previous problem provides an important foundation, so let’s do it.
The prime factors of 324 are: 2 * 2* 3 * 3 * 3 * 3
Exponential notation just groups these as: 2^2 *3^4
The prime factors of 432 in exponential notation are: 2^4 * 3^3
Now, for any number, say x, to be a multiple of 324, it must have factors: (2^2 * 3^4) * a
For any number, say y, to be a multiple of 432, it must have factors: (2^4 * 3^3) * b
The numbers a and b may be small or large, they may even share factors, and that make lots of math work. To find the LCM (least common multiple) of 324 and 432, we want the numbers a and b to include just the factors of the other value that it doesn’t already have. That means that 324 needs another 2^2 and that 432 needs another 3 as factors. Here is our LCM:
a = 2^2
b = 3
LCM = 2^4 * 3^4 (put values of a and b into above expressions)
Note: this LCM is the least value that both 324 and 432 will divide into.
Now, this problem: “find the smallest possible value of a whole number w if 432 x w is a multiple of 324”
This problem says that 432*w is a multiple of 324 and we must find the smallest value of w. Note that we used a and b in part 1, but this question wants us to find what we called “b” to determine the LCM of 324 and 432. Well, we said that b=3 formed the LCM of 432*3, so 3 is the smallest possible whole number that we can multiply 432 by (because the LCM is the “lowest” multiple) to get a multiple of 324.
