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Try this beautiful problem from the Pre-RMO, 2019 based on Sum of digits.
Sum of digits – PRMO 2019
Consider the set E of all natural numbers n such that when divided by 11,12,13 respectively, the remainders, in that order, are distinct prime numbers in an arithmetic progression. If N is the largest number in E, find the sum of digits of N.
- is 107
- is No largest value
- is 840
- cannot be determined from the given information
Key Concepts
Largest Number
Divisibility
Integer
Check the Answer
But try the problem first…
Answer: is No largest value.
Source
Suggested Reading
PRMO, 2019, Question 20
Elementary Number Theory by David Burton
Try with Hints
First hint
here N can be of the form (13)(12)(11)(k)+29
Second Hint
where k belongs to an integer
Final Step
then no largest value.
Other useful links
- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA
Related Program
Math Olympiad Program… taught by Olympians and researchers