Lowest Common Multiples and Highest Common Factors

16-year-old Helena Muffly wrote exactly 100 years ago today: 

Thursday, October 26, 1911: Have such difficult algebra problems. So much work it is to find the H.C. F. and L.C.M. Good bye for me if we happen to get one of these in examination.

Her middle-aged granddaughter’s comments 100 years later:

Important: If you aren’t into math—skip my comments today and come back tomorrow.  Suffice it to say that Grandma was doing some fairly difficult algebra.

But, if you enjoy math here’s my take on what this diary entry is talking about–

First I’ll give an example of the L.C.M. (lowest common multiple) and H.C.F. (highest common factor) of two whole numbers (integers);  then I’ll explain how it’s done for algebraic expressions.

Integers

The L.C.M. is the smallest integer that two whole numbers can be divided by.  1 would always be the L.C.M.

For example, for 8 and 12  the L.C.M. would be 1.

The H.C.F.(highest common factor) is the largest integer that two whole numbers can be divided by.

For the same two numbers (8 and 12), the H.C.F. would be 4.

If the H.C. F. is 1, it is a prime number.

Algebraic Expressions

The basic idea is the same as for algebraic expressions. For example, for H.C.F. of 2ab and 4a2b is 2a.

But it quickly gets complicated. I’m going to give you directions and examples from a 1911 algebra textbook below for H.C.F. [An aside:  If you really want to understand this concept you might find the information on the algebrahelp.com website helpful.]

Now, here are the directions for finding the H.C.F. in  Durrell’s School Algebra (1912):

The method of finding the H.C.F. is to:

Factor the given expressions, if necessary:

Take the H.C.F. of the numerical coefficients:

Annex the literal factors common to all of the expressions, giving to each factor the lowest exponent which it has in any expression.

Ex. 1:  Find the H.C. F. of 6x2y – 12xy2 + 6y3 and 3x2y2 + 9xy3 – 12y4

6x2y – 12xy2 + 6y3 = 6y(x – y)2

3x2y2 + 9xy3 – 12y4 = 3y2(x2 + 3xy – 4y2) = 3y2(x + 4y)(x – y)

H.C.F. = 3y(x – y)

Whew, I’m getting a headache just typing these expressions. But if you’re still with me, here’s a couple problems you could try from the 1911 textbook:

Find the H.C.F.

1. 4a2b , 6ab2

2. x2 – 3x , x2 – 9

3. x2 + x , x2 – 1 , x2 – x – 2

4. 4a3x – 4ax3 , 8a2x3 – 8ax4 , 4a2x2(a – x)2

5. 3a2 – 10a + 3 , 9a – a3 , (3 – a)3

7 Responses

  1. [...] have waited until Sunday evening to get stressed.  In yesterday’s entry she was struggling with L.C.M. (lowest/least common multiples) and H.C.F. (highest common factor) in algebra.  Maybe she still doesn’t get them. Like this:LikeBe the first to like this [...]

  2. Haha…I feel for her. I never enjoyed LCM and HCF math problems either. They always seemed a bit pointless to me….

  3. Heya, Sheryl, above and beyond, that’s what this post was!!

  4. Brenda and Joan–
    I had fun pulling this post together. I’m glad you enjoyed it.

  5. [...] next day.  It seems odd that Grandma doesn’t mention studying for them in this diary entry. The previous Thursday she had mentioned that she was struggling in algebra and worried about upcoming exams. And on [...]

  6. [...] October Grandma struggled mightily with algebra—the topic was least common multiples (L.C.M.) and highest common factors (H.C.F.)—and she ended up getting a 68% on the [...]

  7. [...] Lowest Common Multiples and Highest Common Factors [...]

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