Katagiri calls a number nude if it is divisible by each of its digits
e.g like 672.
Clearly this definition (there exist other as well!) implies non-zero numbers only.
a. How many nude numbers are there below 1,000,000?
b. Evaluate the smallest nude number which contains all the odd digits.
c. Find the smallest triplet of consecutive nontrivial nude numbers.
d. Prove that there is infinite number of nude numbers.
b. 1+3+5+7+9=25 which means increasing the digit sum by 2 or maybe 11.
If we include a 2 this doesn't help as the number would have to end in zero. The smallest would then be 1237950. Better is to use two 1's. I believe the answer is then 1117935.
c. My best is 1111115, 1111116, 1111117. The string of 1's is a must because two consecutive numbers can't be divisible by any digit besides 1.
d. Can be shown in many ways. Any repunit is nude as is any repdigit.
Posted by Jer
on 2018-08-07 09:39:01