# What is meant by GGT

## GCD and LCM

Prime factorization

\ (24 = 2 \ cdot2 \ cdot2 \ cdot3 \)

\ (36 = 2 \ cdot2 \; \; \; \; \; \ cdot3 \ cdot3 \)

GCD & LCM

\ (\ Rightarrow {g} gT (24; 36) = 2 \ cdot2 \ cdot3 = 12 \)

\ (\ Rightarrow {k} gV (24; 36) = 2 \ cdot2 \ cdot2 \ cdot3 \ cdot3 = 72 \)

Shorten and GCT

\ (\ require {cancel} \ frac {24} {36} = \ frac {2 \ cdot2 \ cdot2 \ cdot3 \; \;} {2 \ cdot2 \; \; \ cdot3 \ cdot3} = \ frac {\ cancel {2} ^ 1 \ cdot \ cancel {2} ^ 1 \ cdot2 \ cdot \ cancel {3} ^ 1 \; \;} {\ cancel {2} _1 \ cdot \ cancel {2} _1 \; \; \ cdot \ cancel {3} _1 \ cdot3} = \ frac23 \)

\ (\ require {cancel} \ frac {24} {36} = \ frac {2 \ cdot2 \ cdot3 \ cdot2} {2 \ cdot2 \ cdot3 \ cdot3} = \ frac {12 \ cdot2} {12 \ cdot3} = \ frac {\ cancel {12} ^ 1 \ cdot2} {\ cancel {12} _1 \ cdot3} = \ frac23 \)

Expand and LCM

\ (\ frac1 {24} + \ frac1 {36} \)

\ (= \ frac1 {2 \ cdot2 \ cdot2 \ cdot3} + \ frac1 {2 \ cdot2 \ cdot3 \ cdot3} \)

\ (= \ frac {1 \ color {# EE4D2E} {\ cdot3}} {2 \ cdot2 \ cdot2 \ cdot3 \ color {# EE4D2E} {\ cdot3}} + \ frac {1 \ color {# EE4D2E} {\ cdot2}} {2 \ cdot2 \ color {# EE4D2E} {\ cdot2} \ cdot3 \ cdot3} \)

\ (= \ frac3 {72} + \ frac2 {72} = \ frac5 {72} \)

To shorten the fraction \ (\ frac {24} {36} \), we use the prime factorization of 24 and 36 and quickly see which factors can be shortened: 24 and 36 both contain twice the 2 and a 3 (as Factors) - these can therefore be shortened.

Since nothing else occurs twice, the fraction is then completely shortened. Overall, you can divide both 24 and 36 by twice the 2 and once the 3 whole numbers - the GCF is \ (2 \ cdot2 \ cdot3 = 12 \). Incidentally, you can also see from the prime factorization which numbers remain: Since about \ (24 = 2 \ cdot2 \ cdot2 \ cdot3 \), after shortening the GCF, a 2 remains here - with \ (36 = 2 \ cdot2 \ cdot3 \ cdot3 \) analogously a 3.

Now for expanding and LCM: In order to be able to add fractions, they have to be given the same name by expanding - that is, brought to the same denominator. But what is the best way to expand 24 and 36?

If we use the prime factorization of the two numbers again, we can quickly see which factors are missing in order to get the same denominator: A 3 is missing for 24 and 2 is missing for 36 (as a factor). If we expand the fractions accordingly, we have not only somehow brought them to the kgV 72 with the same name, but optimally with the same name.