# Numerical example for $\gcd(a,b)=\prod p_i^{\min(a_i,b_i)}$

I'm actually having trouble understanding the above corollary. Can anyone please provide a numerical example of that corollary? Thank You So Very Much in advance.

Corollary If $$a=\prod p_i^{a_i}$$ and $$b=\prod p_i^{b_i}$$ where the $$p_i$$ are distinct primes, then $$\gcd(a,b)=\prod p_i^{\min(a_i,b_i)}$$. The LCM is $$\prod p_i^{\max(a_i,b_i)}$$ and the product of the last two expressions is $$ab$$.

Certainly: consider $$a=48=2^4\cdot 3$$ and $$b=40=2^3\cdot 5$$. Then we let $$p_1=2, p_2=3, p_3=5$$ and observe that $$a_1=4, a_2=1, a_3=0$$ and $$b_1=3, b_2=0, b_3=1$$. Your corollary then tells us that \begin{align*} \gcd(48,40) &=\gcd(a,b) \\ &= \prod p_i^{\min(a_i,b_i)} \\ &= p_1^{\min(a_1,b_1)}\cdot p_2^{\min(a_2,b_2)}\cdot p_3^{\min(a_3,b_3)} \\ &= 2^{\min(4,3)}\cdot 3^{\min(1,0)}\cdot 5^{\min(0,1)} \\ &= 2^3\cdot 3^0 \cdot 5^0 \\ &= 8, \end{align*}

Using a similar method we determine that

\begin{align*} lcm(48,40) &=lcm(a,b) \\ &= \prod p_i^{\max(a_i,b_i)} \\ &= 2^4\cdot 3^1 \cdot 5^1 \\ &= 240. \end{align*}

We can then observe that $$ab=48\cdot 40=1920=8\cdot 140=\gcd(48,40)\cdot lcm(48,40)=\gcd(a,b)\cdot lcm(a,b).$$

Let me know if anything needs clarifying (and if anyone knows the command that Latex recognises as a math operator for lcm).

• Formula for LCM should use Max and not minumum. The calculations are right sl – Edcookie274 Feb 17 '19 at 15:23
• Thanks for pointing that out. Fixed. – Ben Feb 17 '19 at 15:32
• Thank You so much!!!! Your answer gave me complete clarity! !!! Thank you once again!!! – Jessi Jha Feb 20 '19 at 3:18

\begin{align}a=600&=2^3\cdot 3^1 \cdot 5^2\\ b=54&=2^1\cdot 3^3 \cdot 5^0\\[2ex] \text{GCD}(a,b)&=2^1\cdot 3^1\cdot 5^0=6\\ \text{LCM}(a,b)&=2^3\cdot 3^3\cdot 5^2=5400\\[2ex] \text{GCD}(a,b)\cdot\text{LCM}(a,b)&=2^4\cdot 3^4\cdot 5^2=32400=ab \end{align}

• Thank You so much!!!! All these examples really helped me!! – Jessi Jha Feb 20 '19 at 3:19

For example, $$540 = 2^2\cdot 3^3\cdot 5$$ and $$72 = 2^3\cdot 3^2$$, so their GCD is $$2^2\cdot 3^2 = 36$$ and their LCM is $$2^3\cdot 3^3\cdot 5 = 1080$$. The point is that if a number $$d$$ is to divide both $$a$$ and $$b$$, the power to which each prime appears in the factorization of $$d$$ cannot exceed the mininum power to which it appears in the factorization of both $$a$$ and $$b$$. A similar analysis works for the LCM.

• Thank You so much!!! I got it! ! – Jessi Jha Feb 20 '19 at 3:21