A 100-digit positive integer is divisible by 72. What is the greatest possible value of the sum of the number’s digits?



Help please


Can you give me step by step or an explanation please

 Feb 17, 2019

((10^100) - 64) mod 72 =0 


9999999999999999999999999 9999999999999999999999999 9999999999999999999999999 9999999999999999999999936 =100 digits long!

Sum =[98 x 9] + 3 + 6 =891

 Feb 17, 2019

what does mod mean? can you explain it please?

IDontKnowMath  Feb 17, 2019
edited by IDontKnowMath  Feb 17, 2019

Mod is short for "Modulus". When you divide a number by the "modulus" it tells you right away if it is divisible by the modulus EVENLY, when the result =0. If it returns any other number, which is the REMAINDER, that means the number does not divide the modulus EVENLY.  Examples: 72 mod 7 =2 The remainder. But 77 mod 7=0, which means it divides the modulus evenly with no remainder.

 Feb 17, 2019

Oh ok thank you soo much!

IDontKnowMath  Feb 17, 2019

Guest, I know what mod is but I do not follow your logic. You lost me right at the beginning.

Where did 64 come from?  

I did get the same asnwer as you, so I guess your logic is sound but I cannot follow it.


Here are my thoughts.


A 100-digit positive integer is divisible by 72. What is the greatest possible value of the sum of the number’s digits?


72=9*8    So it must be divisable by both 9 and by 8


if a number is divisable by 9 the digits have to add to a multiple of 9

1000 is divisable by 8, any number ending in 000 will be divisable by 8.


so the first   97 digits could be 9


now I have to think about the units, 10s and 100 places.


The sum of the 3 digits must be add to 9,18 or 27.    (in order for the number to be disable by 9)

27 is no good because the number would end in 9 and so not be divisable by 8

How about 18     

864  that works    

so I am going to say that the greatest possible value for the sum of the digits is


9*97+18 = 891

 Feb 18, 2019

Hi Melody: He/she wants a 100-digit number which is divisible by 72.

I simply tried: 1E100 mod 72 =64 as the remainder. So, if I subtracted it from 1E100, the number would consist of all 9s and 36 and the end and be divisible by 72 with no remainder, which is the case. At the same time, it gives him/her the largest sum of the 100 digits, 98 of them being 9s and the last two being 36.

 Feb 18, 2019

ok thanks I get it.

You had to use a calc for that of course.   :)


10^100 mod 72 = 64   I just used the web2.0 calc for that.


So the biggest number possible will be  10^100 - 64

the smallest 2 digits will be 36 and all the rest will be 9.  

9*98+9 = 891.    


Ok I can see what you have done and I can see it has worked this time but

I question if it would work everytime.



I am just thinking out loud now.


You have found the biggest number, but I am nost so sure that the biggest number would always have the greatest sum of the digits for a question like this.

I mean I can see that it has to work for this particular example because 

the 2 smallest have to add up to 9,   or

the 3 smallest have to add to 18   or

the 4 smallest have to add to 27 etc

And of course those small numbers can be placed anyway, they do not have to have the smallest place value.


but for other examples.......not so sure.


997>989   however   9+9+7<9+8+9


I do like your logic though - thanks for sharing it :)

Melody  Feb 18, 2019

5 Online Users