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SOD (Posted on 2008-07-09) Difficulty: 3 of 5
Denote "sod(x)" as sum of the digits of x. Examples: sod(49) = 4 + 9 = 13; sod(123) = 1 + 2 + 3 = 6.

Find sod(sod(sod(4444^4444))).

See The Solution Submitted by pcbouhid    
Rating: 4.0000 (1 votes)

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Solution Solution | Comment 1 of 2

At the outset,  let A = sod(4444^4444), B = sod(A), and C = sod(B).  We are required to evaluate C= sod(B).

Now, we have: 4444 (mod 9) = 7
-->   4444^4444(mod 9) = 7^4444

Now, 7^4444 (Mod 9) = ((7^3)^1481)*7 (mod 9) = (1^1481)*7 (Mod 9) = 7

Thus, A leaves a remainder of 7, when divided by 9. Since any number and its sum of digits will leave the same remainder, when divided by 9, it follows that B will also leave a remainder of 7, when divided by 9.

Now, we observe that:

4444^4444 < (100000)^4444 = (10^4)^4444 = 10^(17,776)
-->4444^4444 has less than 17,776 digits.

Therefore, A < 9* 17777 < 199999

So, B < 1 + 5* 9 = 46

But, B must possess the form 7(mod 9). Since B (mod 9) = C, we must have C (mod 9) = 7. If possible, let C >= 16.

Now, the maximum value of C, with 1<= B <= 46 occurs whenever B = 39, giving C = 12 < 16. This is a contradiction.

Since C< 16, and C (mod 9) = 7, it follows that C must be equal to 7.

Consequently,  sod ( sod ( sod (4444 ^ 4444 ) ) ) = 7.

Edited on July 10, 2008, 6:20 am
  Posted by K Sengupta on 2008-07-09 11:37:01

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