Our Target Number: 72
Divide 72 by small primes (2, 3, 5...) until you hit 1. List the ingredients.
Convert your list into Powers. Count how many 2s and how many 3s.
72 is mathematically defined as $2^3 \times 3^2$.
We can systematically build all factors of 72 using a grid of its prime components ($2^3$ and $3^2$).
Examine the grid below carefully.
Q1: Why is $2^0 \times 3^0$ in the table?
Look at the headers again.
Every cell is a factor. Total Factors = Rows $\times$ Columns = 12.
Notice the pattern:
Validation: Why is there a "+1" in the formula $(a+1)(b+1)$?
Total Factors = $(a+1)(b+1)$
To find HCF and LCM, we must check every prime base individually.
A = 24 ($2^3 \times 3^1$) | B = 36 ($2^2 \times 3^2$)
The HCF must fit inside BOTH numbers. We need the Minimum power for each prime.
HCF = $2^2 \times 3^1 = 12$.
Confirmed: It is the Highest Common Factor!
The LCM must be big enough to hold EITHER number. We need the Maximum power for each prime.
LCM = $2^3 \times 3^2 = 8 \times 9 = 72$.
Confirmed: It is the Lowest Common Multiple!