Question 1132815: Rustam likes multiples of 5. For his RSM project he makes up numbers such that any two of the digits of any of his numbers, when put next to each other in some order, make a two-digit multiple of 5. His teacher calls such numbers Rustam’s numbers. For example, the numbers 51 and 502 are Rustam’s numbers, but the numbers 300 and 2018 are not. How many different three-digit Rustam’s numbers with none of the digits greater than 5 are there?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
The problem sounds as if it might be interesting; but the statement of the problem is not clear.
You say 300 and 2018 are not Rustam's numbers; but 30 is a 2-digit multiple of 5, and 10, 20, and 80 are all 2-digit multiples of 5.
There is no point in our trying to count the number of Rustam's numbers if we don't have a clear definition of what they are.
Answer by ikleyn(52787)  (Show Source):
You can put this solution on YOUR website! .
These 3-digit numbers are such that they
- EITHER contain at least two digits 5, (category 1)
- OR contain both the digits 5 and 0, (category 2)
- OR contain two digits 0. (category 3).
In category 1 we have 13 numbers
155, 255, 355, 455, 555,
515, 525, 535, 545,
551, 552, 553, 554.
In category 2 we have 18 numbers
105, 205, 305, 405, 505,
150, 250, 350, 450, 550,
510, 520, 530, 540,
501, 502, 503, 504.
In category 3 we have 5 numbers.
100, 200, 300, 400, 500.
In all, there are 13 + 18 + 5 = 36 such numbers.
It may happen that I missed something, but the idea is clear.
This hint is EITHER the full solution OR a good push for those who study at RSM (Russian School of Mathematics).
P.S. If you think that the numbers of category 3 should not be included to the set of R-numbers
(in this part your definition is not certain), then exclude them . . .
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