Millie wrote a five-digit whole number on a blackboard and she also wrote it in reversed order. She considered the difference of her two numbers, and then told Lucy the last three digits of this difference. Can Lucy figure out the first two digits of the difference from this information alone or there could be more than one possibility?

Question

cloirdxti · Accepted Answer

Step 1Recall the tests for divisibility by 9 and 11:The sum of digits of the number is ≡±od9 to the original numberThe sum/difference of the digits of the number (units minus tens plus hundreds minus thousands plus tens of thousands etc.) is ≡±od{11} to the original number.If the numbers ABCDE― and EDCBA― are written using digits A,B,C,D,E in those two orders, then they are both ≡±od{9} to A+B+C+D+E and they are both ≡±od{11} to A−B+C−D+E. This means that their difference is divisible by 9 and by 11.Thus, if someone has provided last three digits RST― of the difference PQRST―, then the whole difference is given by:PQRST―≡RST―(mod1000)PQRST―≡0(mod9)PQRST―≡0(mod11)As per Chinese Remainder Theorem, this determines PQRST― up to (±od{9⋅11⋅1000}) (as 9, 11 and 1000 are all co&#039;). As this difference is a difference of two 5-digit numbers, it is between 0 and99999−10000=89999so being uniquely determined up to (mod 99000) means it is uniquely determined, period.Answer: yes, knowing the last three digit of the difference, one can recover the first two digits by demanding that the whole 5-digit difference is divisible by 9 and 11.

Millie wrote a five-digit whole number on a

Answered question

Answer & Explanation

New Questions in Pre-Algebra