N is the smallest positive integer such that the sum of the digits of N is 18 and the sum of the digits of 2N is 27. Find N.
Since sod(N) = 18, and sod(2N) = 27, it follows that the first digit from the left must be 9.
If N contains 3 digits, then: the number must be 9--x--y. Then, x+y = 9. Checking for (x y) = (0,9) (1,8), (2,7), (3,6), (4,5), (5,4), (6,3), (8,1), (9,0) it is observed that the identity sod(2N) = 27 is not satisfied in any of these ten cases. This leads to a contradiction.
If N contains 4 digits, then it must possess the form 9-x-a-b. Checking for x =0, we have:
a+b=9, which is not satisfied whenever a= 0 to 9.
If x=1, then a+b= 8
Checking for a= 0 to 9, we ovserve that a=b=4, satisfies the equation: sod(2(9+x+a+b)) = 27, since:
sod(9144*2) = sod(18288) = 1+8+2+8+8 = 27.
A value of N is 9144.
[EDIT]
Now, every permutation of the four digits satisfy the given conditions.
The smallest of these permutations is 1449.
Therefore, the required smallest value of N is 1449.[/EDIT]
Edited on January 16, 2024, 8:53 pm
Puzzle Solution
