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How many digits does 5437682*34567254 have?

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by Max@Math Revolution » Sun Sep 29, 2019 11:19 pm
[GMAT math practice question]

How many digits does 5437682*34567254 have?

A. 13
B. 14
C. 15
D. 16
E. 17
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by GMATGuruNY » Mon Sep 30, 2019 3:00 am
Max@Math Revolution wrote:[GMAT math practice question]

How many digits does 5437682*34567254 have?

A. 13
B. 14
C. 15
D. 16
E. 17
(5*10�)(3*10�) = 15*10¹³ = 150,000,000,000,000 --> 15 digits.
(6*10�)(4*10�) = 24*10¹³ = 240,000,000,000,000 --> 15 digits.
Since these products each have 15 digits -- and 5437682*34567254 is BETWEEN them -- it must also have 15 digits.

The correct answer is C.
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by Max@Math Revolution » Tue Oct 01, 2019 11:45 pm
=>

Assume x = 5437682 and y = 34567254.
We have 5.4 * 10^6 < x < 5.5 * 10^6 and 3.4 * 10^7 < y < 3.5 * 10^7.
Then we have (5.4 * 10^6)(3.4 * 10^7)< xy < (5.5 * 10^6)(3.5 * 10^7) or 18.36*10^{13} < xy < 19.25*10^{13}, which is equivalent to 1.836*10^{14} < xy < 1.925*10^{14}.
So, xy has 15 digits.

Therefore, C is the answer.
Answer: C
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