If 10^a * 3^b * 5^c = 450^n, what is the value of c?

Question

If  10^a * 3^b * 5^c = 450^n , what is the value of c?

(1) a is 1.
(2) b is 2.

Abhishek009 · Accepted Answer

10^a * 3^b * 5^c = 450^n Or, 2^a*3^b*5^{c + a} = 2^n * 3^{2n} * 5^{2n} FROM STATEMENT - I ( INSUFFICIENT ) c + a = 2n Or, c + 1 = 2n ( Not Possible ) FROM STATEMENT - II ( INSUFFICIENT ) If 2n = b = 2 n = 1 c + a = 2 ( Not Possible ) FROM STATEMENT I & II ( INSUFFICIENT) c = 2n - 1 = 2 - a Or, c = 2n + a = 1 Hence, Correct answer will be (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed. PS: Is it me alone or is there someone who feels that this question must be in the DS forum ??? :roll:

Bunuel · Answer

The correct answer is E here because we don't know whether c and n are positive integers or not.

msk0657 · Answer

5^a * 2^a * 3^b * 5^c = 5^2n * 3^2n * 2^n ( 450 = 15 * 6 * 5 )

5^a+c = 5^2n => a+c = 2n --1

2^a = 2^n => a = n --2

3^b = 3^2n => b = 2n --3..

Stat 1 : a = 1.. then

From eq 2 : a = n i.e. n =1.. sub this value in eq 1 we get c = 1...Sufficient.

Stat 2: b = 2n , given b = 2. then n = 2 in eq 1 we get c =1...Sufficient.

IMO D is correct answer...

georgekaterji · Answer

I think something is wrong with OA no?

From statement 1, it is sufficient to get the value of n, same goes for statement 2:S

GMATinsight · Answer

10^a * 3^b * 5^c = 450^n 
i.e.  2^a * 3^b * 5^{(a+c)} = 2^n * 3^{2n} * 5^{2n} 
i.e (a+c) = 2n and a=n and b=2n
i.e. c = 2n-n= n

Question : c = ? i.e. n=?

Statement 1 : a is 1 
i.e. n = 1 = c
SUFFICIENT

Statement 2 : b is 2 
i.e. b = 2n = 2
i.e. n = 1 = c
SUFFICIENT

Answer: option D

GMATinsight · Answer

Bunuel  Could you please post the official solution of this question???

Bunuel · Answer

We are given  10^a * 3^b * 5^c = 450^n .

For (1)+(2) we know that a = 1 and b = 2 -->  10 * 3^2 * 5^c = 450^n . Now, for any n, there would exist corresponding c (not necessarily 1) which would make this equation hold. For example, if n = 2, then c = (log(2)+2 log(3)+3 log(5))/(log(5)) ≈ 4.7959

sai897 · Answer

The correct answer is E here because we don't know whether c and n are positive integers or not.

Bunuel  Could you please post the official solution of this question???

We are given  10^a * 3^b * 5^c = 450^n .

For (1)+(2) we know that a = 1 and b = 2 -->  10 * 3^2 * 5^c = 450^n . Now, for any n, there would exist corresponding c (not necessarily 1) which would make this equation hold. For example, if n = 2, then c = (log(2)+2 log(3)+3 log(5))/(log(5)) ≈ 4.7959

How did you get the log expression from the actual one.. I'm confused

Bunuel · Answer

Bunuel  Could you please post the official solution of this question???

We are given  10^a * 3^b * 5^c = 450^n .

For (1)+(2) we know that a = 1 and b = 2 -->  10 * 3^2 * 5^c = 450^n . Now, for any n, there would exist corresponding c (not necessarily 1) which would make this equation hold. For example, if n = 2, then c = (log(2)+2 log(3)+3 log(5))/(log(5)) ≈ 4.7959

How did you get the log expression from the actual one.. I'm confused

It does not matter at all. The point is that for for any n, there would exist corresponding c (not necessarily 1) which would make this equation hold.

TheUltimateWinner · Answer

GMATinsight , How did you get the red part brother? In red part, if you say a=n, then i'll say a=2n. Then the result will be like below. -->c = 2n-2n -->c=0 Without statement 1 and 2, you already get your value C=0........interesting!

ak1802 · Answer

Don't like this question one bit, but here are the flawed assumptions we're all making:

Assumption 1) that c and n are integers 
Assumption 2) that c and n are positive

Given both statement 1 and statement 2, we are likely to conclude that c = 1, implying n = 1. However, in the unlikely, but possible event that n or c are negative or not integers, we are incorrect.

I prefer the scenario where n is 2 and c is ~4.795889.

This was pointed out earlier using the log function. Since we cannot arrive at a single answer choice, the answer is E.

This problem is to demonstrate integer constraints, not necessarily algebra.

Posted from my mobile device

colorblind · Answer

Good one.
Take an example:  2^c = 825 
If 'c' is an integer, there will be no value for 'c' which will satisfy the above equation 
But, if 'c' is not an integer, there will always be a value for 'c' - a power - which will satisfy the equation given above.

2^c = 825

c = \frac{log 825}{log 2}

c = ~9.68825
Hence 2^(9.68825) = 824.99

Hence both statements together are not enough.

Shiv2016 · Answer

I solved it this way and selected D which is not the correct answer.

2^a * 5^(a+c) * 3^b = (2^1 *5^2 *3^2)^n
2^a * 5^(a+c) * 3^b= 2^n * 5^2n * 3^2n

So I inferred that:
1) a=n
2) a+c= 2n
3) b=2n

So c=n

Skywalker18 · Answer

Prime factorization of 450 = 2* 3^2 * 5^2
450^n = 2^n * 3^2n * 5^2n
Since 10^a * 3^b * 5^c = 450^n
=> 10^a * 3^b * 5^c = 2^n * 3^2n * 5^2n
=> 2^a * 3^b * 5^(a+c) = 2^n * 3^2n * 5^2n

(1) a is 1.
Equating the powers of 2 on both sides
we get a=n=1
Now equating the powers of 5 on both sides ,
a+c = 2n
=> 1+c = 2*1 = 2
=> c = 1

If n = 1. When Statement 1 gives you that a = 1, 
we get 10^1 * 3^b * 5^c = 450^1
=>3^b * 5^c = 45
Value of c depends on value of b . b need not be an integer here .
Not sufficient

(2) b is 2. 
Equating the powers of 3 on both sides
2= 2n
=> n = 1
If n = 1, a = 1 .
Now equating the power of 5 on both sides ,
a + c = 2n 
=>1 + c = 2
=>c = 1

10^a * 5^c = 50
However , as in statement 1 , c can take infinite values depending on value of a .

Not sufficient
Answer E

lakshya14 · Answer

If we were given "n" is positive, then were we good to go with (D)?

If 10^a * 3^b * 5^c = 450^n, what is the value of c?

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GMAT Data Sufficiency (DS) Questions

If 10^a * 3^b * 5^c = 450^n, what is the value of c?

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