What is the least integer z for which (0.000125)(0.0025)(0.00000125)

Question

What is the least integer z for which  (0.000125)(0.0025)(0.00000125)*10^z  is an integer?

A. 18
B. 10
C. 0
D. −10
E. −18

(PS00774)

Princ · Accepted Answer

OA: A
 (0.000125)(0.0025)(0.00000125)*10^z= 125*10^{-6}*25*10^{-4}*125*10^{-8}*10^z 
 =125*25*125*10^{-6-4-8+z} 
 =125*25*125*10^{-18+z}

125*25*125  will end with  5  as unit's digit, there will be no  0  at unit's place.So for  z  to be minimum, -18+z  should be equal to  0 .
 -18+z=0 
 z=18

KSBGC · Answer

This question needs no calculation. Just think.

If we use negative exponent as a value of z we won't get integer in this case. Eliminate D and E. For the same reason o can't be the value of z as we need integer.

We are left with A and B. If anyone calculate the fraction he gets 18 decimal points. A is the best answer.

The best answer is A.

EgmatQuantExpert · Answer

Solution

Given:  
 • We are given an expression:  (0.000125)(0.0025)(0.00000125)∗10^z

To find:  
 • We need to find the least integer z such that  (0.000125)(0.0025)(0.00000125)∗10^z  is an integer

Approach and Working:

•  (0.000125)(0.0025)(0.00000125)∗10^z  can be written as  (125* 10^{-6})(25*10^{-4})(125* 10^{-8})* 10^z 
• =  125*25*125* 10^{-6-4-8+z} 
• For  125*25*125* 10^{-6-4-8+z}  to be an integer, the power of 10 in  10^{-6-4-8+z}  must be an integer, which is greater than or equal to 0
 o Implies, -6 - 4 - 8 + z ≥ 0
o z ≥ 18

Hence, the correct answer is option A.

Answer: A

sudarshan22 · Answer

= 125 ∗ 25 ∗ 125 ∗ 10^(−6−4−8+z)
= 125 ∗ 25 ∗ 125 ∗ 10^(−18+z)

For the above value to be integer, we need a a value of Z that equals 0, therefore 18
-18 + z = 0
-18 + 18 = 0

Hence,   A  .

Abhishek009 · Answer

(0.000125)(0.0025)(0.00000125)*10^z

= (\frac{125}{10^6})(\frac{25}{10^4})(\frac{125}{10^8})10^z

= (\frac{5^3}{10^6})(\frac{5^2}{10^4})(\frac{5^3}{10^8})10^z

= \frac{5^8}{10^{18}}10^z

Now, if  z = 18 , then we will have an Integer with value  5^8 , Answer must be (A)

ScottTargetTestPrep · Answer

0.000125 = 5^3 x 10^-6

0.0025 = 5^2 x 10^-4

0.00000125 = 5^3 x 10^-8

So the product equals 5^8 x 10^-18 x 10^z. We see that if z = 18, then the product will be an integer.

Answer: A

Target4bschool · Answer

The whole expression can be boiled down to: 
5^8 * 10^(-18 +z)

Now, to make this an integer, we need to multiply by 10, enough to eliminate the decimals. 
So, the minimum number of 10s we need are 18 (because we have 10^-18 in the expression)

If z=18, then the expression becomes 5^8 * 10^0 = 5^8 which is an integer

Answer: option A

SURBHIKOTHS · Answer

But for the example: 0.24 x 0.25 x 0.25 the product is : 0.015. By the approach above the value of z here should be 6, but the answer is actually 3, because 0.24 is even

Posted from my mobile device

Target4bschool · Answer

SURBHIKOTHS Good question! In the original question, we have only 5^x And we know that only 5^x will not yield 10 Now, in your question, there are factors of 10 0.24 x 0.25 x 0.25 = (8*0.03) *(5*5* 0.01)*(5*0.05) = (8*5*5*5) * 0.03 * 0.01 * 0.05 = (10^3) * 15*10^(-6) So, to neutralize 3-6 =-3, we need 3. Hope it helps. Let me know if you have further questions. Please give kudos, it motivates

Nsp10 · Answer

If we include 11 in the option, then can we say the least Integer value is 11 ?

Bunuel · Answer

No. As shown in multiple posts above, (0.000125)(0.0025)(0.00000125) * 10^z = 5^8 * 10^(z-18). This expression will be an integer for any integer value of z that is 18 or more. So, the least value of z is 18.

Nsp10 · Answer

The overall expression is 
5^8 * 10^z (Numerator)
___________
10^18 (Denominator)

so we can further simplify it as

(5/10)^8 * 10^z-10 to be integer 
so 0.5*10^z-10
so if z=11 the overall value will be 5 which is integer.

I am sorry but am I wrong anywhere ?

Bunuel · Answer

The red part is wrong. (5/10)^8 does not equal 0.5. Here how t should be:

\frac{5^8 * 10^z}{10^{18}} = 5^8 * 10^{z-18}

Nsp10 · Answer

Ohh I am sorry ,
got it
I totally forgot that will be 0.5^8.
Thank you Legend

EnriqueDandolo · Answer

Hey  Bunuel , I hope you are doing fine. I would like to ask you if you could please explain how can I answer the question if it were phrased like this:

"What is the least integer z for which 0.25 * 0.25* 0.12 *10^z is an integer?"

My question comes from the fact that in the original question I just equaled "z" to the number of times I moved the decimal point to the right in order for each number to be an integer. However that actually wouldnt work in the hypothetical question that I wrote, as the answer would be z=4 instead of z=6 as my erroneous method of solving would suggest.

Thanks in advance for your help.

Bunuel · Answer

In the original question, we can represent each fraction as a product of some power of 10 and some power of 5:

(0.000125)(0.0025)(0.00000125)*10^z=(10^{-6}*5^3)(10^{-4}*5^2)(10^{-8})5^3)* 10^z  .

Here, 5^3 * 5^2 * 5^3 does not produce a trailing zero, so we have:

(10^{-6}*5^3)(10^{-4}*5^2)(10^{-8}*5^3)* 10^z=10^{-18}*5^8*10^z .

The above implies that the least integer value of z such that the whole expression is an integer is 18.

For your example, we'd have:

(0.25)(0.25)(0.12) * 10^z =(10^{-2} * 5^2)(10^{-2} * 5^2)(10^{-2} * 12) * 10^z =10^{-6} * 5^2 * 5^2 * 12 *10^z

Here, since 12 contains 2^2, it would pair with 5^2 and produce two trailing zeros:

10^{-6} * 5^2 * 5^2 * 12 *10^z= 10^{-6} * 10^2 * 5^2 * 3 *10^z= 10^{-4} * 5^2 * 3 *10^z

The above implies that the least integer value of z such that the whole expression is an integer is 4.

Hope it's clear.

totaltestprepNick · Answer

A https://www.youtube.com/watch?v=f7IdbymAGl4 Nick Slavkovich, GMAT/GRE tutor with 20+ years of experience tutoring@totaltestprep.net

Usernamevisible · Answer

Doubt - can I not take 100 from 125, 10 from 25 and while arriving at z giving these multiplication will reduce the requirement of power of 10, reduce z

Key insight:
Your idea cancels powers of 5 (like taking 100 or 10),
but the limiting factor is powers of 2.

Product:
= (125 * 25 * 125) / 10^18
= (125^2 * 25) / 10^18

Factor:
125 = 5^3
25 = 5^2
⇒ numerator = 5^8

10^18 = (2 * 5)^18 = 2^18 * 5^18

So:
= 5^8 / (2^18 * 5^18)
= 1 / (2^18 * 5^10)

Multiply by 10^z:
= (2^z * 5^z) / (2^18 * 5^10)
= 2^(z−18) * 5^(z−10)

Denominator has 2^18, so we must have:
z ≥ 18

Also:
z ≥ 10 (from 5s)

Final:
Minimum z = 18

What is the least integer z for which (0.000125)(0.0025)(0.00000125)

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GMAT Problem Solving (PS) Questions

What is the least integer z for which (0.000125)(0.0025)(0.00000125)

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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