Common, Multiply Me

Number Theory Level 2

The least common multiple of $2 \times 3 \times 5 \text{ and } N$ is $2 \times 3^2 \times 5 \times 7$ . What is the smallest possible value of $N$ ?

The answer is 63.

5 solutions

Nandan Agarwal
Oct 2, 2013

As we are given that 2 X 3 X 5 and N has LCM as 2 x $3^{2}$ x 5 x 7 and LCM means highest powers of factors in all the 4 number since $3^{2}$ and 7 does not appear in prime factorization of 2, 3, 5 therefore N= $3^{2}$ X 7 = 63

Moderator note:

Great explanation of finding $N$ from the factors that we require.

nice logic used

Devesh Rai - 7 years, 8 months ago

You can also use the identity $\gcd(a,b)*\text{lcm}(a,b)=ab$ .

Arkan Megraoui - 7 years, 6 months ago

Daniel Ferreira
Sep 30, 2013

Para encontrarmos o MMC entre dois números fatorizados: - pegamos os fatores incomuns; - e, os maiores expoentes dos fatores comuns.

Como o enunciado pede que calculemos o menor valor possível,...

$\begin{cases} 2 \times 3 \times 5 \\ 3^2 \times 7 \end{cases} \\ -------- \\ 2 \times 3^2 \times 5 \times 7$

Daí,

$N = 3^2 \times 7 \\\\ \boxed{N = 63}$

Niky Wulandari
Sep 29, 2013

N= 3^2 x 7 = 63

You need to explain more about how you got to the fact that 3^2*7=63

Mohith Manohara - 7 years, 8 months ago

This however means that the number they give you is smaller than 2x3x5xN which is not a least common multiple.

Vishnu Tanguturi - 7 years, 8 months ago

I don't agree with this. If you look again at the first equation, its product will be of 30N. If we are going with N = 63, then it'll be 1890. The problem: The least common multiple of 2×3×5 and N is 2×3^2×5×7. Which means the least is 630; three times smaller than 1890.

So by going with this logic, N should be of N = 630/30, which is 21. Not 63.

Ervina Wijaya - 7 years, 8 months ago

but, axb=gcd(a,b)x lcm(a,b)
gcd(30,63)=3. so, 30xN=30x63=1890 and gcd(a,b) x lcm(a,b) = 3x630=1890. So, the solution is correct.

Tamoghna Mukherjee - 7 years, 8 months ago

The answer is correct since it is the LCM of 2 different digits (2 3 5) and N. Therefore,it is the LCM of 30 and N..

neha adepu - 7 years, 8 months ago

I think so

Niky Wulandari - 7 years, 8 months ago

Kevin Hanafi
Mar 30, 2014

the least common multiply is 2 x 3^2 x 5 x 7 one of the number is 2 x 3 x 5 In finding least common multiply, we must use all of the numbers with the biggest power or degree So, the smallest number of N is 3^2 x 7

Gaurang Pansare
Oct 6, 2013

The least common multiple is $2\times 3^{2}\times 5\times 7$

Now therefore,

N has to be a multiple of 7
the fact that 3 is squared implies that N has to be a multiple of 9

Therefore, the smallest possible value of N is 9x7=63

Common, Multiply Me

The answer is 63.

5 solutions

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