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Bunuel
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If a and b are integers, and 240a = b 3 , which of the following must 17 Feb 2020, 08:39
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58% (01:58) correct 42% (02:24) wrong based on 315 sessions

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If \(a\) and \(b\) are integers, and \(240a = b^3\), which of the following must be an integer?

I. \(\frac{a}{300}\)

II. \(\frac{a}{600}\)

III.\(\frac{a}{450}\)

A. None
B. I only
C. III only
D. I and III only
E. I, II, and II


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D
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If a and b are integers, and 240a = b 3 , which of the following must 17 Feb 2020, 12:39
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If a and b are integers, and \(240a=b^{3}\), which of the following must be an integer?

\(240a=2^{4}*3*5*a= b^{3}\\
\)

In order b to be the integer, a must be at least \(2^{2}* 3^{2}*5^{2}= 4*9*25=900\)

I. \(\frac{a}{300}\)--YES

II. \(\frac{a}{600}\)- NO

III. \(\frac{a}{450}\) -YES

The answer is D.
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If a and b are integers, and 240a = b 3 , which of the following must 17 Feb 2020, 15:11
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If a and b are integers, and 240a = b 3 , which of the following must 18 Feb 2020, 03:09
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240 = 2^4*3*5 and a has to be 2^2*3^2*5^2 for both LHS = RHS
for given option only option 1 & 3 are true and would give an integer value
IMO D


Bunuel
If \(a\) and \(b\) are integers, and \(240a = b^3\), which of the following must be an integer?

I. \(\frac{a}{300}\)

II. \(\frac{a}{600}\)

III.\(\frac{a}{450}\)

A. None
B. I only
C. III only
D. I and III only
E. I, II, and II


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If a and b are integers, and 240a = b 3 , which of the following must 30 Mar 2020, 09:13
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Bunuel
If \(a\) and \(b\) are integers, and \(240a = b^3\), which of the following must be an integer?

I. \(\frac{a}{300}\)

II. \(\frac{a}{600}\)

III.\(\frac{a}{450}\)

A. None
B. I only
C. III only
D. I and III only
E. I, II, and II
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\(240a = b^3…24*10a=b^3…2^4*3*5*a=b^3\)
\(2^4*3*5*a=perfect.cube…(powers.primes.must.be.multiples.of.3)\)
\(perfect.cube=min(2^6*3^3*5^3)\)
\(min(a)=2^2*3^2*5^2\)

\(a/pf(300)=3*100=3*2^2*5^2…must.be.div\)
\(a/pf(600)=6*100=3*2^3*5^2…min(a)=2^2≠2^3…may.not.be.div\)
\(a/pf(450)=45*10=3^2*2*5^2…must.be.div\)

Ans (D)
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If a and b are integers, and 240a = b 3 , which of the following must 20 Jan 2022, 22:45
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why cant we assume a and b to be 0?
In this case 240 (0)= (0)^3 and A,B,C will not be an integer so answer is none
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If a and b are integers, and 240a = b 3 , which of the following must 21 Jan 2022, 01:47
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riarr
why cant we assume a and b to be 0?
In this case 240 (0)= (0)^3 and A,B,C will not be an integer so answer is none
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I think you are missing that zero is an integer.

ZERO:

1. Zero is an INTEGER.

2. Zero is an EVEN integer.

3. Zero is neither positive nor negative (the only one of this kind)

4. Zero is divisible by EVERY integer except 0 itself (\(\frac{x}{0} = 0\), so 0 is a divisible by every number, x).

5. Zero is a multiple of EVERY integer (\(x*0 = 0\), so 0 is a multiple of any number, x)

6. Zero is NOT a prime number (neither is 1 by the way; the smallest prime number is 2).

7. Division by zero is NOT allowed: anything/0 is undefined.

8. Any non-zero number to the power of 0 equals 1 (\(x^0 = 1\))

9. \(0^0\) case is NOT tested on the GMAT.

10. If the exponent n is positive (n > 0), \(0^n = 0\).

11. If the exponent n is negative (n < 0), \(0^n\) is undefined, because \(0^{negative}=0^n=\frac{1}{0^{(-n)}} = \frac{1}{0}\), which is undefined. You CANNOT take 0 to the negative power.

12. \(0! = 1! = 1\).



2. Properties of Integers



For more check Ultimate GMAT Quantitative Megathread



Hope it helps.
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If a and b are integers, and 240a = b 3 , which of the following must 21 Jul 2025, 01:38
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