A 10 digit positive integer is called a
cute number if its digits are from the set {1,2,3} and every two consecutive digits differ by one.
a. Prove that exactly five digits of a cute number are equal to 2.
b. Find the total number of cute numbers.
c. Prove that the sum of all cute numbers is divisible by 1408.
Source: Romanian math competition
For the third part:
Consider the number N=2222222222
For each cute number M>N whose difference to N is A, it does exists another cute number P<N whose difference to N is A, and viceversa.
So the sum of all cute numbers is:
N*64=142222222222=1408*101010101.
Edited on February 2, 2018, 6:12 am

Posted by armando
on 20180202 05:30:17 