Find lcm(143, 227), lcm(306, 657), etc.? Can anyone verify my work?

[Math100]

Homework Statement

Find lcm(143, 227), lcm(306, 657), and lcm(272, 1479).

Relevant Equations

None.

Proof: First, we will find lcm(143, 227).
Note that lcm(a, b)=abs(a*b)/[gcd(a, b)].
Now we will find gcd(143, 227).
Applying the Euclidean Algorithm produces:
227=1(143)+84
143=1(84)+59
84=1(59)+25
59=2(25)+9
25=2(9)+7
9=1(7)+2
7=3(2)+1
2=2(1)+0.
Thus, gcd(143, 227)=1.
Since gcd(143, 227)=1, it follows that lcm(143, 227)=abs(143*227)/[gcd(143, 227)]=32461.
Therefore, lcm(143, 227)=32461.
Next, we will find lcm(306, 657).
Note that lcm(a, b)=abs(a*b)/[gcd(a, b)].
Now we will find gcd(306, 657).
Applying the Euclidean Algorithm produces:
657=2(306)+42
306=7(42)+12
42=3(12)+6
12=2(6)+0.
Thus, gcd(306, 657)=6.
Since gcd(306, 657)=6, it follows that lcm(306, 657)=abs(306*657)/[gcd(306, 657)]=33507.
Therefore, lcm(306, 657)=33507.
Finally, we will find lcm(272, 1479).
Note that lcm(a, b)=abs(a*b)/[gcd(a, b)].
Now we will find gcd(272, 1479).
Applying the Euclidean Algorithm produces:
1479=5(272)+119
272=2(119)+34
119=3(34)+17
34=2(17)+0.
Thus, gcd(272, 1479)=17.
Since gcd(272, 1479)=17, it follows that lcm(272, 1479)=abs(272*1479)/[gcd(272, 1479)]=23664.
Therefore, lcm(272, 1479)=23664.

 

[Math100]

gcd(306, 657) correct. The other two look good.

So everything is correct/looks good?

 

[phyzguy]

No.

 

[phyzguy]

gcd(306,657) is not 6.

 

[phyzguy]

Sorry, my first post was garbled.

 

[Math100]

I see now, I made mistakes for the second subproof. It should be the case that gcd(306, 657)=9. Am I right?

 

[phyzguy]

Yes

 

[Math100]

Thank you so much for the help!