Runirish wrote:

If m is a positive integer, then m^3 has how many digits?

1. m has 3 digits

2. m^2 has 5 digits

How would you do this quickly? Is there a rule that I am unaware of? I could do it, but I had to pick a few numbers. Thanks.

1. m has 3 digits

When I look at such statements, I invariably think of the extremities. (as muralimba did above)

Smallest m = 100 which implies m^3 = 10^6 giving 7 digits.

Largest m = 999 but it is not easy to find its cube so I take a number close to it i.e. 1000 and find its cube which is 10^9 i.e. smallest 10 digit number. Hence 999^3 will have 9 digits.

Since we can have 7, 8 or 9 digits, this statement is not sufficient.

2. m^2 has 5 digits

Now try to forget what you read above. Just focus on this statement.

Smallest m^2 = 10000 which implies m = 100

Largest m^2 is less than 99999 which gives m as something above 300 but less than 400.

Now, if m is 100, m^3 = 10^6 giving 7 digits.

If m is 300, m^3 = 27000000 giving 8 digits.

Since we have 7 or 8 digits for m, this statement is not sufficient.

Now combining both, remember one important point -

If one statement is already included in the other, and the more informative statement is not sufficient alone, both statements will definitely not be sufficient together.e.g. statement 1 tells us that m has 3 digits. Statement 2 tells us that m is between 100 and 300 something, so statement 2 tells us that m has 3 digits (what statement 1 told us) and something extra (that its value lies between 100 and 300 something). Statement 2 is more informative and is not sufficient alone. Since statement 1 doesn't add any new information to statement 2, no way will they both together be sufficient.

Hence answer (E).

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