Problems & Puzzles: Puzzles

Puzzle 548.- dr(p*n) = dr(n)

Farideh Firoozbakht sent the following nice puzzle.

For sure you know that dr(n) is the digital root of n
One of my friends - Jahangger Kholdi - has found that : 19 is the smallest prime p such that for each positive integer n, dr(p*n) = dr(n).
Q1. Can you prove it ?
Q2. Prove that dr(m*n) = dr(dr(m)*dr(n))  and so deduce that all numbers of the form 9k+1 have the property that Jahangeer has mentioned for 19.
Q3. Prove that dr(m+n) = dr(dr(m)+dr(n)).



Contributions came from Jan van Delden & Luke Pebody


Jan wrote:

Since dr(n )=1+((n-1) mod 9 ) statements Q2 and Q3 follow immediately (and hence Q1) from:
ab mod c = (a mod c * b mod c) mod c and
a+b mod c = (a mod c + b mod c) mod c
If only equivalence is necessary the last mod c on the right side can be removed.


Luke wrote:

dr(n) = n mod 9. Thus dr(m)=dr(n) for distinct positive integers m,
n if and only if 9 is a factor of m-n.

Since dr(n)=dr(dr(n)) (clearly), this means that 9 is a a factor of
n-dr(n) for all n.

Thus 9 is a factor of (m - dr(m)) + (n - dr(n)) = (m + n) -
(dr(m)+dr(n)), so dr(m+n) = dr(dr(m)+dr(n)).
Furthermore, 9 is a factor of n(m - dr(m)) + dr(m)(n - dr(n)) = nm -
ndr(m) + ndr(m) - dr(m)dr(n) = nm-dr(m)dr(n), so



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