MHB Calculating LCM with Multiple Prime Factors

[karush]

$\tiny{c4.LCM of \{A,B,C\}}$

Spoiler

added image to avoid typo

Build the LCM of $\{A,B,C\}$
Then write the prime factorization of the LCM of $\{A,B,C\}$.
$A = 2 \cdot 3 \cdot 7^2 \cdot 19$
$B = 2^2 \cdot 17^9 \cdot 19^8 \cdot 23^8$
$C = 2^6 \cdot 3 \cdot 11^9 \cdot 19$
$LCM of \{A,B,C\}=\boxed{?}$

ok well to start with $\{A,B,C\}$. all have a common factor of 2 and 19
the lowest of 2 is 2 and the lowest of 19 is 19

so LCM $\{A,B,C\}$.=(2)(3)(7^2)(17^9)(19)(23^8)

 

[HOI]

NO! For "least common multiple" you take the HIGHEST power of each prime, not the lowest.

 

[karush]

write the prime factorization of the LCM of $\{A,B,C\}$.
$A = 2 \cdot 3 \cdot 7^2 \cdot 19$
$B = 2^2 \cdot 17^9 \cdot 19^8 \cdot 23^8$
$C = 2^6 \cdot 3 \cdot 11^9 \cdot 19$

so LCM $\{A,B,C\}$ = (2^6)(3)(7^2)(17^9)(19^8)(23^8)

hopefully

however only 2 and 19 are only in all 3 sets?

 

[HOI]

Yes, so what is your question?

If you were asked to find the least common multiple of 2, 3, and 5, you should respond with 2(3)(5)= 30, even though NONE of those factors are in all 3 numbers.

 

[karush]

ok I don't see 5 in there

NO! For "least common multiple" you take the HIGHEST power of each prime, not the lowest.

write the prime factorization of the LCM of $\{A,B,C\}$.
$A = 2 \cdot 3 \cdot 7^2 \cdot 19$
$B = 2^2 \cdot 17^9 \cdot 19^8 \cdot 23^8$
$C = 2^6 \cdot 3 \cdot 11^9 \cdot 19$

so then LCM $\{A,B,C\} = (2^6)(3)(7^2)(11^9)(17^9)(19^8)(23^8)$
)

 

[sumaira1]

Lcm is the least common factor the number which is multiples of all numbers for example Lcm of 3 and 4 is 12
the other method is prime factorization.

Step 1: Express each number as a product of prime factors.

Step 2: LCM = The product of the highest powers of all prime factors.Step 1 : Express each number as a product of prime factors.

18 = 2 × 32

24 = 23 × 3

9 = 32

36 = 23 × 32

90 = 2 × 5 × 32

Step 2: LCM = The product of the highest powers of all prime factors.

Here the prime factors are 2, 3 and 5

The highest power of 2 here = 23

The highest power of 3 here = 32

The highest power of 5 here = 5

Hence LCM = 23 × 32 × 5 = 360

 

[HOI]

sumaira, good post but it would be easier to read if you were to use "^" to indicate exponents.

Lcm is the least common factor the number which is multiples of all numbers for example Lcm of 3 and 4 is 12
the other method is prime factorization.

Step 1: Express each number as a product of prime factors.

Step 2: LCM = The product of the highest powers of all prime factors.Step 1 : Express each number as a product of prime factors.

18 = 2 × 32

18= 2 x 3^2 so I won't think that is "thirty two"!

24 = 23 × 3

24= 2^3 x 3

9 = 32

9= 3^2

36 = 23 × 32

36= 2^3 x 3^2

90 = 2 × 5 × 32

90= 2 x 5 x 3^2

Step 2: LCM = The product of the highest powers of all prime factors.

Here the prime factors are 2, 3 and 5

The highest power of 2 here = 23

2^3

The highest power of 3 here = 32

3^2

The highest power of 5 here = 5

Hence LCM = 23 × 32 × 5 = 360

2^2 x 3^2 x 5= 360
(23 x 32 x 5= 3680!)

 

[sumaira1]

yeah i understand next time I will use this notation for power