GCF and LCM of 30 and 40?

Answer 1

Instead of just giving you the answer as I did for your other question, let me explain what GCF and LCM are, and give you a general method for finding them.

The GCF, or greatest common factor of two numbers, is the largest integer which is a factor of both numbers - that is, both numbers can be divided by it with no remainder. There are three general methods for finding the GCF:

1: guess and check

To find the GCF of two numbers A and B, start with the lower of the two numbers (let's assume it's B), and check all the numbers B, B/2, B/3, B/4... 1 to determine whether they are integers, and if they are, whether they are factors of A. The first number tested that meets both of these criteria is the GCF. This method is the simplest, but also by far the slowest, unless you get lucky and the GCF is one of the first few numbers tested.

2. prime factor decomposition

Decompose both A and B into products of prime factors (this step is not trivial). Then find all of the prime factors that appear in BOTH the prime decompositions of A and B, and if any factors appear multiple times in either product, write down that factor the same number of times that it appears in the product that contains the least number of that factor. Then find the product of these numbers to get the GCF of A and B. For instance, if your numbers are 728 and 196, your prime decompositions would be (2*2*2*7*13) and (2*2*7*7), thus your GCF would be (2*2*7), which is 28. The most time-consuming step in this method is finding the prime factors, however, many of the numbers you work with in math class and in real life have obvious prime factorizations, so when working with small numbers (2-3 digits), this is often the best method in practice.

3: Euclid's method

For finding the GCF of large numbers, neither of the above methods are usually sufficient. Fortunately, there is a faster method. Let A mod B be the remainder of A divided by B - e.g. 9 mod 4=1. Then we may find the GCF of A and B as follows: find A mod B. If the answer is zero, then B is the GCF and we are done. Otherwise, find the GCF of B and A mod B. Repeat this procedure until you get zero. The last number you divided by is your GCF.

Example: finding the GCF of 78920 and 65109.
78920 mod 65109=13811
65109 mod 13811=9865
13811 mod 9865=3946
9865 mod 3946=1973
3946 mod 1973=0

Thus the GCF of 78920 and 65109 is 1973.

Why this works: the Euclidian algorithm is based on the observation that if two numbers have a common factor, then the remainder of these numbers after division also has that factor. This is easily shown as follows: if q is the quotient of A/B and r is the remainder, then A = qB + r, with q and r integers. Divide both sides by f: A/f = qB/f + r/f and thus r/f = A/f - qB/f. Suppose f is a factor of both A and B: then A/f and B/f are also integers, and as q is an integer, qB/f is also an integer. Since the difference of two integers is also an integer, r/f is an integer and thus f is a factor of r. Q.E.D.

The LCM, or least common multiple, is the smallest integer that both numbers are a factor of - that is, the LCM can be divided by either of them with no remainder. Finding the LCM is pretty simple once you know how to find the greatest common factor: simply divide the product of A and B by the greatest common factor of A and B. That is: LCM(A, B) = A*B/GCF(A, B).

Answer 2

GCF=10
LCM=120

Answer 3

Prime factorization yields...

30 = 10 * 3 = 5 * 2 * 3

40 = 10 + 4 = 5 * 2 * 2 * 2

To find the GCF, simply group the common prime factors...

5*2 is common to both, so th GCF = 10.

To find the lcm, we must be a bit more clever, but can still use number theory.

Simply multiply both prime factorizations by new terms until they match each other...

Thus, 5 * 2 * 3 = 5 * 2 * 2 * 2?

The LCM is clearly 5 * 2 * 2 * 2 * 3 = 120.

Answer 4

Lcm Of 30 And 40

Answer 5

gcf = 10
lcm = 120

Answer 6

gcf=10
lcm=120

Answer 7

10 and 120

Answer 8

the greatest common factor is 10. the least common multiple is 120.
prime factorization of 30= 2x3x5
" " " " " 40=2x2x2x5
2x2x2x3x5=120, if im not mistaken...

Answer 9

You'd be much better off asking HOW rather that WHAT'S THE ANSWER...


:-)