MHB Finding 3-Digit Numbers with Sum of Digits Squared = 2

AI Thread Summary
The discussion focuses on identifying 3-digit numbers where the sum of their digits squared equals 2. It highlights that valid examples include 110 and 101, both of which meet the criteria. The conversation also touches on the two-digit numbers 20 and 11, questioning if 200 qualifies as a 3-digit number with a digit sum of 20. Ultimately, the correct total of such 3-digit numbers is confirmed to be 85. The participants acknowledge the findings and clarify the categories of solutions.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
For any natural number $n$, let $S(n)$ denote the sum of the digits of $n$. Find the number of all 3-digit numbers $n$ such that $S(S(n))=2$.
 
Mathematics news on Phys.org
First, 110 and 101 are three digit numbers with digits sum 2 and then its digit sum is again 2. (I assume you would not consider "011" a three digit number.)

Now the only two digit numbers with digit sum 2 are 20 and 11. What three digit numbers have digit sum 20 or 11?
 
Not 200?

-Dan
 
Sum of all the digits of a 3 digit number that is $S(n)$ shall be between 1 and 27.because $(S(n)) = 2 $ and $S(n)$ is between 1 and 27 so we have $(n) \in \{ 2,11,20\}$ Now the number of 2 digit numbers(with leading zero) sum = n is n+1 for n <=9

This is so because 1st digit can be 0 to n and 2nd digit can be The number of 1/2 digit numbers sum = n is 19-n for 9 < n < 19 is 19-n. This is so because the 1st digit can go from n -9 to 9 or 9-(n-9) +1 =19-n numbersOne digit number is allowed ( that is 2 digit number with leading zero) because we are considering the tens digit of the 3 digit number that can be zero.Now we shall use the above 2 do the countingThe 3 digit numbers that have sum 3 are 101,110,200 that is 3 numberslet us count the number of 3 digit that have a sum 111st digit is 1 so sum of other 2 digit 10 so number of numbers = 19-10 = 91st digit is 2 to 9 so sum of other 2 digit 9 down to 2 number of numbers = $10 + 9 + \cdots 3 = \frac{(10 + 3)*8}{2} = 52$So the number of numbers that have sum 11 is 9 + 52 = 61let us count the number of 3 digit that have a sum 201st digit can be from 2 to 9 and that shall give the sum of other 2 digits of the number can be from 18 down to 11. the number of numbers can be 19-k with k from 18 to 11so the sum = $1+2 + \cdots 8 = \frac{8 * 9}{2} = 36$So total number of numbers with sum 20 = 36So total number of numbers = 3 + 61+ 36 = 100
 
Sorry all, the correct answer is 85. (Nod)
 
****************
checking for 2
****************
1 : The number is 101
2 : The number is 110
3 : The number is 200
****************
checking for 11
****************
1 : The number is 119
2 : The number is 128
3 : The number is 137
4 : The number is 146
5 : The number is 155
6 : The number is 164
7 : The number is 173
8 : The number is 182
9 : The number is 191
10 : The number is 209
11 : The number is 218
12 : The number is 227
13 : The number is 236
14 : The number is 245
15 : The number is 254
16 : The number is 263
17 : The number is 272
18 : The number is 281
19 : The number is 290
20 : The number is 308
21 : The number is 317
22 : The number is 326
23 : The number is 335
24 : The number is 344
25 : The number is 353
26 : The number is 362
27 : The number is 371
28 : The number is 380
29 : The number is 407
30 : The number is 416
31 : The number is 425
32 : The number is 434
33 : The number is 443
34 : The number is 452
35 : The number is 461
36 : The number is 470
37 : The number is 506
38 : The number is 515
39 : The number is 524
40 : The number is 533
41 : The number is 542
42 : The number is 551
43 : The number is 560
44 : The number is 605
45 : The number is 614
46 : The number is 623
47 : The number is 632
48 : The number is 641
49 : The number is 650
50 : The number is 704
51 : The number is 713
52 : The number is 722
53 : The number is 731
54 : The number is 740
55 : The number is 803
56 : The number is 812
57 : The number is 821
58 : The number is 830
59 : The number is 902
60 : The number is 911
61 : The number is 920
****************
checking for 20
****************
1 : The number is 299
2 : The number is 389
3 : The number is 398
4 : The number is 479
5 : The number is 488
6 : The number is 497
7 : The number is 569
8 : The number is 578
9 : The number is 587
10 : The number is 596
11 : The number is 659
12 : The number is 668
13 : The number is 677
14 : The number is 686
15 : The number is 695
16 : The number is 749
17 : The number is 758
18 : The number is 767
19 : The number is 776
20 : The number is 785
21 : The number is 794
22 : The number is 839
23 : The number is 848
24 : The number is 857
25 : The number is 866
26 : The number is 875
27 : The number is 884
28 : The number is 893
29 : The number is 929
30 : The number is 938
31 : The number is 947
32 : The number is 956
33 : The number is 965
34 : The number is 974
35 : The number is 983
36 : The number is 992

In the above I find 100 solutions specified under the categories. $S(n)$
 
Last edited:
Ah...you're right, kaliprasad! (Nod) And thanks for your participation!
 
[
For 3 digit number the lowest value of S(n) is when n is 100 that is S(100) = 1 and largest when n = 999 that is S(999) = 27
Now to calculate S(n) when n is 1/2 digit number <= 27 is 1 to 10 10 is when n = 19 otherwise it is a single digir number

so $S(S(n))$ for 3 digit number is between 1 to 10

Further n- S(S(n)) is divsible by 9.

so n has a remainder $S(S(n))$ when divided by 9 and $S(S(n))$ is not 9 or 10

n has a remainder $S(S(n))-9$ when divided by 9 and $S(S(n))) is 9 or 10

as S(S(n) is 2 so n has a remainder 2 and the numbers from 11 * 9 + 2 to 110 * 9 +2 that is 100 3 digit numbers satisfy the criteria
 

Similar threads

Replies
3
Views
2K
Replies
2
Views
2K
Replies
11
Views
2K
Replies
1
Views
912
Replies
4
Views
2K
Replies
2
Views
909
Replies
1
Views
1K
Back
Top