If a=2b=6c for positive integers a, b, and c, which of the following

Question

If  a=2b=6c  for positive integers  a, b,  and  c , which of the following could be the value of  abc ?

A. 1000
B. 1230
C. 2250
D. 2367
E. 2488

charlie-23 · Answer

a = 2b
2b = 6c ---> b = 3c

abc = 2b*b*c = 2*3*3*c*c*c = 18*c^3

For any positive integer c, abc must be a multiple of 18.

It should be multiple of 2, 9
For 2, it must be even.
For 9, sum of all digits should be multiple of 9.
Those conditions are satisfied in C.

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Paras96 · Answer

If a=2b=6c for positive integers a,b, and c , which of the following could be the value of abc?

abc = (6c)(3c)c = 18c^3

A. 1000 => Not a multiple of 18 hence rejected 
B. 1230=> Not a multiple of 18 hence rejected  
 C. 2250 Answer  
 D. 2367=> Not a multiple of 18 hence rejected 
E. 2488=> Not a multiple of 18 hence rejected

Henrymavill · Answer

Approach I followed is this:

A=2b I.e. b= A/2

Similiarly, C= A/6

Abc = A^3/12 which implies that for Abc to be an integer, it has to be divisible by 12 and none of the said options satisfy this, can someone tell me where I went wrong?

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Flower_Child · Answer

Hi Henrymavill, Your approach is fine but your inference is wrong.

In  Abc = A^3/12   , for abc to be an integer it is not necessary for ABC to be divisible by 12, instead  A^3  should be divisible by 12.
Again , we cannot comment anything about the individual A, B, C values also .

To correct this, we need to convert the equation in a form that removes the denominator.
We know that  A^3 = 8 B^3 and , A^3 =216 C^3

Getting rid of the denominator, the eq becomes , 
 ABC= A^3/12 = \frac{216 C^3 }{ 12 } 
 ABC = 18 C^3

Thus we have to check the divisibility of 2 and 9 only.

bumpbot · Answer

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If a=2b=6c for positive integers a, b, and c, which of the following

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If a=2b=6c for positive integers a, b, and c, which of the following

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