Answer:
- 2
- 2
Explanation:
You want the remainder of a number when divided by 3 if (1) the sum of digits is 5, and (2) the remainder from division by 9 is 5.
(1) Digit sum is 5
The sum of digits of a 2-digit number will be 5 if that number is one of ...
{14, 23, 32, 41, 50}
In every case, the remainder when divided by 3 is 2. Effectively, it is the remainder of 5 when that is divided by 2.
(2) Mod 9 value is 5
In addition to the numbers listed above, 2-digit numbers with a sum of digits of 14 will also have a remainder of 5 when divided by 9. This adds five more numbers to the list:
{59, 68, 77, 86, 95} . . . . have remainders of 5 when divided by 9
The remainders when divided by 3 are all 2 for these numbers as well.
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Additional comment
The process of (recursively) summing the digits of a positive integer is called "casting out 9s". It effectively finds the modulo 9 value of the number, with the exception that a sum of 9 corresponds to a modulo 9 value of 0.
Since 9 is divisible by 3, the modulo 3 value of a number will be the modulo 3 value of the modulo 9 value. In other words, if the sum of digits is 5, or the remainder from division by 9 is 5, then the mod 3 value is 5 mod 3 = 2.