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# If A is an integer, what is the value of A?

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Math Expert
Joined: 02 Sep 2009
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If A is an integer, what is the value of A?  [#permalink]

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30 Apr 2020, 02:29
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35% (medium)

Question Stats:

68% (01:56) correct 33% (01:49) wrong based on 40 sessions

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If A is an integer, what is the value of A?

(1) $$1,000,000 < 23^A < 10,000,000$$

(2) $$\frac{7^{A+2}−7^A}{48}=7^A$$

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If A is an integer, what is the value of A?  [#permalink]

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30 Apr 2020, 02:41
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Bunuel wrote:
If A is an integer, what is the value of A?

(1) $$1,000,000 < 23^A < 10,000,000$$

(2) $$\frac{7^{A+2}−7^A}{48}=7^A$$

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Question: A = ?

Statement 1: $$1,000,000 < 23^A < 10,000,000$$

$$23^2 = 529$$

$$(23^2)^2 = 529^2 ≈ 500^2 ≈ 250,000$$ i.e. Less than 1,000,000

$$23^5 = (23^2)^2*23 ≈ 250000*23 ≈ 6,250,000$$
i.e. between 1,000,000 and 10,000,000

$$23^6 ≈ 6250000*23 ≈ 14,000,000$$ i.e. Greater than 10,000,000

i.e. A = 5

SUFFICIENT

Statement 2: $$\frac{7^{A+2}−7^A}{48}=7^A$$

i.e. $$7^A*(7^2-1) = 7^A*48$$

i.e. 48 = 48, No value fo A hence

NOT SUFFICIENT

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Re: If A is an integer, what is the value of A?  [#permalink]

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30 Apr 2020, 03:00
1
Bunuel wrote:
If A is an integer, what is the value of A?

(1) $$1,000,000 < 23^A < 10,000,000$$

(2) $$\frac{7^{A+2}−7^A}{48}=7^A$$

Project PS Butler

Are You Up For the Challenge: 700 Level Questions

(Statement1): $$10^{6} < 23^{A} < 10^{7}$$

$$23^{2}= 529$$

$$23^{4}= 529*529 > \frac{10^{3}}{2}* \frac{10^{3}}{2}= \frac{10^{6}}{4} < 10^{6}$$

$$23^{5} ≈ \frac{10^{6}}{4}* 23= 5.75* 10^{6} > 10^{6}$$

$$23^{6} ≈ 5.75* 10^{6}* 23= 5.75*2.3* 10^{7} > 10^{7}$$

Well, we got only one value of A satisfying the inequality
A=5
Sufficient

(Statement2): $$\frac{7^{A+2}−7^A}{48}=7^A$$

$$\frac{7^{A} (49 -1)}{48}= 7^{A}$$

--> $$7^{A} = 7^{A}$$
A could be any integer
Insufficient

Re: If A is an integer, what is the value of A?   [#permalink] 30 Apr 2020, 03:00