This was to Jane, who has since deleted her post

I don't see any nonsense in ZHero's post. It may not be as formal as you'd like, but everything in there is correct. That method also gives a lower maximum.

Side note: The smallest number with 45 factors is 3600. Also, any number with that amount of factors must be a square.

To find the remainder we simply need to find the last 3 digits of 9*5^43.  First find the last 3 digits of 5^43, which is fairly simple when you notice the following pattern:

X             Last 3 digits of 5^X
1             005
2             025
3             125
4             625
5             125
6             625
...            ...

For any even number n >= 4, the last 3 digits of 5^n are 625.  For any odd number k >= 3, the last 3 digits of 5^k are 125.  Since 43 is odd we know the last 3 digits of 5^43 are 125.  Multiply this by 9 to get 1125.  This means our remainder when dividing by 1000 will be 125.

The following table gives the details of the ages and heights of the boys and girls attending a hobby camp. The height (in cm) of each boy/girl is an integer.



Age Group                                        Boys                                            Girls

(Age(A),in years)
   
A<=10                                       8(120,130,150)                            10(110,130,140)

11<=A<=13                             10(130,150,160)                              8(120,140,150)

14<=A<=16                               6(150,160,180)                              6(140,150,170)

A>16                                          5(170,180,200)                             8(150,160,180)


In the above table, the number mentioned outside the brackets in each cell gives the number of boys (or girls) in that age group and the values mentioned inside the brackets give the minimum height, the average height and the maximum height of the persons in that group, in that order.


For example, of the eight boys who are of age not more than 10 years, the minimum height of any boy is 120 cm, the average height of all the boys is 130 cm and the maximum height of any boy is 150 cm.

1)A team is to be selected from the boys in the camp who were of age 13 years or below. If only boys who are 140 cm or more in height can be selected, the number of boys who can be selected is at most.

2)If a team of girls of age 13 years or less is to be selected, with the condition that all the girls who are 130 cm or more in height must be selected, what is the minimum number of girls selected?


Please give your approach while giving the solution.

Last edited by Sudeep (2008-08-06 11:49:59)

55! is the product of 55 numbers: 1, 2, 3, ..., 54 and 55.
All of these numbers are less than 100, and so 55! < 100^55 = 10^110.

So 55! has no more than 110 digits. The possible sum of its digits is maximum when they are all 9, and so it maximum possible sum-of-digits is 990.

It's easy to see that the number with the highest sum-of-digits that is not higher than 990, is 899 with a digit-sum of 26.

Similarly, the number between 1 and 26 with the highest digit-sum is 19, which has a sum of 10.
Therefore, the sum of the sum of the sum of the digits of 55! is bounded above by 10.

The divisibility-by-9 test says that a number is divisible by 9 iff its digits sum to a multiple of 9.
55! has 9 as a factor, and so the sum of its digits (and so, the sum-of-sum and the sum-of-sum-of-sum) will also be a multiple of 9.

This reduces the possibilities for the sum-of-sum-of-sum to either 0 or 9.
Clearly 55! has at least one non-zero digit, and so its digit sum will be more than 0. Hence its sum-of-sum and sum-of-sum-of-sum will also be non-zero, which leaves 9 as the only possibility.

The sum of the sum of the sum of the digits of 55! is 9.

Thanks!

I cheated a bit though, at first I used MathsIsFun's full precision calculator to display 55!, then worked out the answer by literally adding all the digits, then the digits of the answer and so on.

I didn't like that method much, so I made the more analytical one after knowing what the answer was.

How many natural numbers having at most six digits have the sum of their digits as at most six?

In a certain system product of 44*11 is represented s 2124 find how 1453 can be represented in decimal system ?

Er … I think Sudeep wrote:

10^5

So you need to include 5-digit numbers as well.

Well, you’ve done a good job so far; I’ll finish it for you.

The number of 5-digit numbers ending ___00 is equal to the combined total number of 2- and 3-digit numbers, namely 63.

For the 5-digt numbers not ending ___00 …

Group 1: 100__

Group 2: 101__, 110__, 200__

Group 3: 102__, 111__, 120__, 201__, 210__, 300__

Group 4: 103__, 112__, 121__, 130__, 202__, 211__, 220__, 301__, 310__, 400__

Group 5: 104__, 113__, 122__, 131__, 140__, 203__, 212__, 221__, 230__, 302__, 311__, 320__, 401__, 410__, 500__

Group 6: 105__, 114__, 123__, 132__, 141__, 150__, 204__, 213__, 222__, 231__, 240__, 303__, 312__, 321__, 330__, 402__, 411__, 420__, 501__, 510__, 600__

Group 7: 106__, 115__, 124__, 133__, 142__, 151__, 160__, 205__, 214__, 223__, 232__, 241__, 250__, 304__, 313__, 322__, 331__, 340__, 403__, 412__, 421__, 430__, 502__, 511__, 520__, 601__, 610__, 700__

Group 8: 107__, 116__, 125__, 134__, 143__, 152__, 161__, 170__, 206__, 215__, 224__, 233__, 242__, 251__, 260__, 305__, 314__, 323__, 332__, 341__, 350__, 404__, 413__, 422__, 431__, 440__, 503__, 512__, 521__, 530__, 602__, 611__, 620__, 701__, 710__, 800__

Group 9: 108__, 117__, 126__, 135__, 144__, 153__, 162__, 171__, 180__, 207__, 216__, 225__, 234__, 243__, 252__, 261__, 270__, 306__, 315__, 324__, 333__, 342__, 351__, 360__, 405__, 414__, 423__, 432__, 441__, 450__, 504__, 513__, 522__, 531__, 540__, 603__, 612__, 621__, 630__, 702__, 711__, 720__, 801__, 810__, 900__

So the total number of 5-digit numbers not ending in two zeros is 1×10 + 3×9 + 6×8 + 10×7 + 15×6 + 21×5 + 28×4 + 36×3 + 45×2 = 660.

Hence total number of natual numbers less than 100000 with digits adding up to 10 is 282 + 63 + 660 = 1005.

I'm pretty sure that the amount of _ _ _00 numbers is the same as the amount of just the 3-digit numbers, so your answer should be 9 less.

I did this a different way. I might've made it overcomplicated though.

Look at this length-10 line:
         ----------

By placing exactly 4 markers along it, you can represent every number less than 10^5 whose digits add up to 10.

eg.  23131 =   --'---'-'---'-

In theory, you can use a combinatorial formula to work out the total number of marker placements, and hence the total amount of numbers with the required property.

However, we get into trouble with this method because combinatorics doesn't work when two markers can be placed on the same point.
According to our symbolism, 12070 would be -'--''-------', but this isn't a valid combination.

Time for a workaround!

Now, we consider a length-15 line and place markers along it, and each line length means one more than the corresponding digit.

eg.  23131 = ---'----'--'----'--
      12070 = --'---'-'--------'-

Now 0 digits can be represented by the system, and so the representation works.

There are 14 places to put the markers (15 lines, so 14 gaps between them) and 4 markers, so our answer is 14C4 = 1001.

...Nearly.

One final tweak is needed, because this representation will also throw out numbers like (10)0000.
Luckily, it's easy to count those (there are 5) so after ignoring them, the final answer is 996.

M=10^5(m5)+10^4(m4)+10^3(m3)+10^2(m2)+10^1(m1)+10^0(m0) ≤ 10^5
⇒m5=0
⇒∑(10^i)mi ≤ 10^5 (i=1,2,3,4)...(1)
∑mi=10....(2) So the inequalility holds as max. value of M is 91000
∴No of M's satisfying (1)&(2) is = coefficient of x^10 in (x^0+x^1+x^2+.....upto ∞ terms )^5 - (5 c 4) (as mi can't take value 10;0≤mi≤10;0<x<1)
                                             =      ,,         ,,   ,,     ,, (1-x)^(-5) - (5 c 4)=(14 c 4) - 5=1001-5=996

Last edited by Subhamoy (2009-05-03 15:54:55)

Hi anonimnystefy,