Question

Find Greatest Common Divisor of 48, 72, 108, 120, 144, 180, 192, 216, 240 and 300, using prime factorization.

Answer

The GCD of given numbers is 12.

Explanation

Step 1 : Find prime factorization of each number.

48 = 72 = 108 = 120 = 144 = 180 = 192 = 216 = 240 = 300 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 2 \cdot 2 \cdot 3 \cdot 5 \cdot 5

(view steps on how to factor 48, 72, 108, 120, 144, 180, 192, 216, 240 and 300. )

Step 2 : Put a box around factors that are common for all numbers:

48 = 72 = 108 = 120 = 144 = 180 = 192 = 216 = 240 = 300 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 2 \cdot 2 \cdot 3 \cdot 5 \cdot 5

Step 3 : Multiply the boxed numbers together:

GC D = 2 \cdot 2 \cdot 3 = 12

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