Playing with Numbers - Common Factors and Common Multiples

Factors and Multiples: A factor of a number is an exact divisor that leaves no remainder. Visualize factors as all possible rectangular arrangements of a set of items; for instance, $12$ items can be arranged in $1 \times 12, 2 \times 6,$ or $3 \times 4$ grids. A multiple is a number obtained by multiplying a given number by any whole number, which can be visualized as equal-sized jumps along a number line.

Common Factors: When two or more numbers have the same factors, those shared values are called common factors. If you represent the factors of two numbers using a Venn Diagram (two overlapping circles), the common factors are the numbers listed in the intersection where the circles overlap.

Highest Common Factor (HCF): The HCF is the largest value among all common factors of two or more numbers. It is also known as the Greatest Common Divisor (GCD). Imagine you have two ribbons of different lengths; the HCF is the length of the longest possible piece you can cut so that both ribbons are divided into equal parts with no leftovers.

Common Multiples: These are numbers that are multiples of all the given numbers in a set. On a number line where you mark multiples of $4$ in red and multiples of $6$ in blue, the common multiples are the points where both red and blue marks appear ($12, 24, 36, \dots$).

Lowest Common Multiple (LCM): The LCM is the smallest non-zero common multiple of two or more numbers. Visualize two runners on a circular track; if one finishes a lap in $40$ seconds and another in $60$ seconds, the LCM ($120$ seconds) is the first time they will both cross the starting line together again.

Co-prime Numbers: Two numbers are called co-prime if their only common factor is $1$. For example, $8$ and $15$ are co-prime because the factors of $8$ are $\{1, 2, 4, 8\}$ and the factors of $15$ are $\{1, 3, 5, 15\}$. In a Venn diagram of their factors, the overlapping section would contain only the number $1$.

Prime Factorization Method: This is a way to find HCF and LCM by breaking numbers down into products of prime numbers. For HCF, we multiply the common prime factors with the lowest powers. For LCM, we multiply all prime factors using their highest powers found in any of the numbers.