SOLUTION: What is the sum of the digits of the product of 23 ones and 32 tens?

Question 996686: What is the sum of the digits of the product of 23 ones and 32 tens?
Answer by KMST(5345) (Show Source): You can put this solution on YOUR website!
MULTIPLYING:
You get , and
adding the digits you get .

WITHOUT MULTIPLYING:
,
so the product is tens,
with the same digits as with a added to the right.
We need to find the sum of the digits of .
If you add the digits of a number, and add the digits of the sum, and so on, until you get a 1-digit number, the final result is the remainder of dividing the original number by 9.
Since ,
when we divide by , the remainder is , and
when we divide by , the remainder is .
When you multiply numbers, the remainder from dividing the product by 9
is the the remainder of dividing the product of the remainders of dividing the factors by 9.
So, the remainder from dividing the product by 9 is
the remainder of dividing by 9.
which is .
However, is number ending in , and
Is a 3-digit number between and .
So, the sum of the first and last digits of that product is greater than , and
the sum of all 3 digits of must be less than .
Then, the sum of all 3 digits of must be a 2-digit number between and ,
The only such number that divided by 9 has a remainder of is
,
because .

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