LCM (least common multiple)

LCM is an abbreviation for Least Common Multiple’. It is also called as ‘lowest common multiple’ and less popularly as ‘smallest common multiple’. For a given set of two or more integers, the least common multiple is the least integer that can be divided by all the integers in the given set.

The fundamental method of finding the smallest common multiple for a set of integers is first list out the multiples of each integer. Pick up the common integers that appear in all such lists. Now figure out the lowest among them. It is the least common multiple. For example, let us try to find the least common multiple of 4 and 6.
The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 …
The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42 …
The integers that are seen common in both the lists are: 12, 24, 36 …
The smallest integer among these commons is 12.
Therefore, the smallest common multiple of 4 and 6 is 12.

There is another method to find the lowest common multiple by using the following formula.
LCM of x, y = (x*y)/(GCF of x and y)
Let us try this method to find the least common multiple of 12 and 16.
The GCF of 12 and 16 is 4.
Therefore, the least common multiple of 12 and 16 = (12*16)/(4) = 3*16 = 48.

In cases where you need to find the lowest common multiples of many numbers, especially large, you can find that by prime factorization. The method here is to do the prime factorization of each number. The lowest common multiple is the product of the highest power of each prime number, out of all factors. Let us find the least common multiple of 9 and 12 by this method.
The prime factorization of 9 is 3*3 = 32.
The prime factorization of 12 is 3*4 =3*2*2 = 3* 22.
The product of the highest powers of prime numbers is 32* 22= 9*4 = 36.
Hence, the smallest common multiple of 9 and 12 is 36.

The knowledge of lowest common multiple for a given set of integers are greatly helpful in finding lowest common denominator for a given set of fractions. The lowest common denominator is the lowest common multiple of the denominators of the given fractions. Therefore, LCM problems are mostly found in such applications. For example the lowest common divisor of the fractions (1/3) and (1/5) is 15, which is the lowest common multiple of 3 and 5.

Definition of consecutive interior angles

Define consecutive interior angles:
Consider two lines intercepted by a transversal as shown in the figure below:

In the above figure, two lines l and m are intercepted by a transversal n. That leads to the formation of four interior angles named a, b, c and d. Of these four angles, angles a and b are called consecutive interior angles and the angles c and d are another pair of consecutive interior angles. In other words, the angles formed on the same side of the transversal are called consecutive interior angles.

Consecutive interior angles examples:

In the above figure following are pairs of consecutive interior angles:
1 – 4, 2 – 3 , 5 – 6, 7 – 8, 9 – 11, 10 – 12, 13 – 16, 14 – 17 etc.

Theorem for consecutive interior angles for parallel lines:
If a pair of parallel lines are intercepted by a transversal, then each of the pairs of consecutive interior angles is supplementary.
See the following figure:

In the above figure, we have two lines l and m that are parallel to each other. The pair of consecutive interior angles marked in blue in this case would add up to 180 degrees or the angles are said to be supplementary to each other.

Consecutive interior angles converse theorem:
If two lines are intercepted by a transversal such that each pair of consecutive interior angles is supplementary ( that means that each pair of alternate interior angle measures add up to 180 degrees) then the lines have to be parallel to each other. See figure below to understand that better:

In the above figure, one of the pair of consecutive interior angles is