For example, the LCM of 10, 24. In factoring the two numbers, there are 4 twos, one 3, and one 5. But, in using compact or exponents to find the LCM, the correct form is 2 cube, multiplied by 3, multiplied by 5, which is 120. However, I get confused because there are 4 twos in factoring both numbers. So, why is it not 2 to the fourth power instead of 2 cube?
Asked
Active
Viewed 11 times
0
-
This is hard to follow. If $a=\prod p_i^{a_i}$ and $b=\prod p_i^{b_i}$ then $\text {lcm}(a,b)=\prod p_i^{\max (a_i,b_i)}$. Nothing is "left out". – lulu Nov 18 '22 at 16:21
-
In your example: $10=2\times 5$ and $24=2^3\times 3$ so $\text {lcm}(10,24)=2^3\times 3\times 5=120$. – lulu Nov 18 '22 at 16:22
-
See the linked proofs, in particular $,p^j,p^k\mid p^i \iff j,k\le i \iff \max(j,k)\le i\ \ $ – Bill Dubuque Nov 18 '22 at 16:24