Eminit
Problem
The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?
Student Solutions
Correct solutions were received form Robert of Forres Academy, Andrei of School NO. 205, Bucharest and Chong, Chen and Teo of Secondary 1B, River Valley High School, Singapore.
Each solution identified that groups of six 8s and six 9s are
divisible by seven.
(Is it the case that all six figure numbers
made by repeating the same digit are divisible by 7?). This
meant that because the original number had 50 digits on each side
of M: there are eight groups of six and two digits left.
Which puts the problem into the form:
What is M if 88M99 is divisible by 7?
Robert and Andrei found M by trial and improvement methods and the full solution from the Valley High School Singapore is given below. Well done
The value of the digit M is 5.
When 888888 is divided by 7, we will get 126984. Besides, 8, 88, 888, 8888 and 88888 cannot be divided 7 without any remainder and 888888 is the smallest number made up of eights which can be divided by 7. Thus, when the digit 8 is repeated 50 times (88888888888888 888888888888888888888888888888888888) is divided by 7, we will get the number 126984 repeated 8 times and 12 behind the number, which is 12698412698412698412698412698412698412698412698412 and has a remainder 4.
On the other hand, when 999999 is divided by 7, we will get 142857. Besides, 9, 99, 999, 9999 and 99999 cannot be divided 7 without any remainder and 999999 is the smallest number made up of nines which can be divided by 7. Thus when the digit 9 is repeated 50 times (99999999 999999999999999999999999999999999999999999) is divided by 7, we will get the number 142857 repeated 8 times and 14 behind the number, which is 142857142857142857142857142857142 85714285714285714 and has a remainder 1.
Thus, when the digit 8 is repeated 50 times and divided by 7, the remainder 4 is carried forward and in order to avoid getting any remainder when the digit 9 is repeated 50 times and is divided by 7, we have to take out the first two 9s. Therefore, 4M99 must be able to be divided by 7, so after many trials and errors, we managed to find out that the number that can be divided by 7, is 4599. Therefore, we are able to conclude that the value of the digit M is 5.
Well done to all of you.