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## Number properties

tagged by: rolandprowess

This topic has 1 expert reply and 0 member replies

### Top Member

Roland2rule Legendary Member
Joined
30 Aug 2017
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#### Number properties

Sat Sep 30, 2017 7:24 pm
s(n) is a n-digit number formed by attaching the first n perfect squares, in order, into one integer. For example, s(1) = 1, s(2) = 14, s(3) = 149, s(4) = 14916, s(5) = 1491625, etc. How many digits are in s(99)?

A. 350
B. 353
C. 354
D. 356
E. 357
What is the best way to dismantle this problem?

### GMAT/MBA Expert

EconomistGMATTutor GMAT Instructor
Joined
04 Oct 2017
Posted:
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Tue Oct 10, 2017 6:01 am
Hello Roland.

In this question, you have to see how many digits have a perfect square when the bases are n=1, 2, 3,....., 99.

1^2=1 has 1 digit
2^2=4 has 1 digit
3^2=9 has 1 digit
4^2=16 has two digits
. . .
Then we only have to add all this results.

But, this is a too large list to make. So, we need a simplification.

You can see that:

- if the base is in the set A={1,2,3}: the square has 1 digit.
- if the base is in the set B={4,5,6,7,8,9}: the square has 2 digits.
- if the base is in the set C={10,11,. . . , 31}: the square has 3 digits.
- if the base is in the set D={32,33, . . . , 99}= the square has 4 digits.

Now, we know that in set A there are 3 numbers, in the set B there are 6 numbers, in C there are 22 numbers and in D there are 68 numbers.

So, s(99) has 3*1+6*2+22*3+68*4=353 digits, where the first factor represents the total of numbers in each set and the second factor represents the total of digits that the square has.

So, the correct answer here is B.

I am available if you would like any follow up.

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