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# Let f(x) denotes the sum of the digits of the positive integer x. For

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Math Expert
Joined: 02 Sep 2009
Posts: 55670
Let f(x) denotes the sum of the digits of the positive integer x. For  [#permalink]

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03 Aug 2018, 03:58
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Difficulty:

95% (hard)

Question Stats:

43% (02:27) correct 57% (02:04) wrong based on 99 sessions

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Let f(x) denotes the sum of the digits of the positive integer x. For example, f(8) =8, f(123) = 1+2+3=6. For how many two-digit values of x is f(f(x))=3?

A. 3
B. 4
C. 6
D. 9
E. 10

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Senior Manager
Joined: 13 Feb 2018
Posts: 329
GMAT 1: 640 Q48 V28
Re: Let f(x) denotes the sum of the digits of the positive integer x. For  [#permalink]

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03 Aug 2018, 04:16
We need two digit integers with the sum of digits equal to 3
here they are
12, 21
30,
49, 94
58, 85
67, 76

IMO
Ans: D
Intern
Joined: 26 Feb 2017
Posts: 24
Re: Let f(x) denotes the sum of the digits of the positive integer x. For  [#permalink]

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03 Aug 2018, 04:21
12,21,30
39,48,57,66,75,84,93

Total 10 possibilities.

So E

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Re: Let f(x) denotes the sum of the digits of the positive integer x. For  [#permalink]

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03 Aug 2018, 04:28
Bunuel wrote:
Let f(x) denotes the sum of the digits of the positive integer x. For example, f(8) =8, f(123) = 1+2+3=6. For how many two-digit values of x is f(f(x))=3?

A. 3
B. 4
C. 6
D. 9
E. 10

f(12)=1+2=3; so, f(f(12))=f(3)=3
f(21)=2+1=3; so, f(f(21))=f(3)=3
f(30)=3+0=3; so, f(f(30))=f(3)=3

Now we have to check any pair of integers those sum to 12, 12, and 31 as f(12) yields 3 and f(f(3))=3.

4+8=8+4=12
5+7=7+5=12
6+6=12
3+9=9+3=12
hence, we have 3+7=10 nos. of two-digit values. (12,21,30,48,84,57,75,66,39,93)

Ans. (E)
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Re: Let f(x) denotes the sum of the digits of the positive integer x. For  [#permalink]

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24 Oct 2018, 22:36
LevanKhukhunashvili wrote:
We need two digit integers with the sum of digits equal to 3
here they are
12, 21
30,
49, 94
58, 85
67, 76

IMO
Ans: D

it will be
12,21,
30,
39, 93
48,84
57, 75
66
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Joined: 16 Oct 2010
Posts: 9330
Location: Pune, India
Re: Let f(x) denotes the sum of the digits of the positive integer x. For  [#permalink]

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25 Oct 2018, 00:34
Bunuel wrote:
Let f(x) denotes the sum of the digits of the positive integer x. For example, f(8) =8, f(123) = 1+2+3=6. For how many two-digit values of x is f(f(x))=3?

A. 3
B. 4
C. 6
D. 9
E. 10

Here is how I will think about it:

x needs to be a 2 digit number. It can be anything from 10 to 99 (minimum sum of digits 1 and maximum sum of digits is 18)
So f(x) will vary from 1 to 18.

f(f(x) = 3 when f(x) = 3 or 12 (note that f(x) cannot be 21 or 30 since f(x) cannot take these values)

f(x) = 3 when x = 30/12/21
f(x) = 12 when x = 39/93/48/84/57/75/66

In all, we have 10 values for x.
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Karishma
Veritas Prep GMAT Instructor

Re: Let f(x) denotes the sum of the digits of the positive integer x. For   [#permalink] 25 Oct 2018, 00:34
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