UNO High School Problem of the Week Competition
Problem 8
Find a multiple of 11 that leaves a remainder of 1 when divided by each of the integers 2,3,5, and 7.
Solution
First lets recognize that (number - 1) is divisble by 2,3,5,7. Therefor are (number - 1) must be divisble by 2*3*5*7 = 210. Therefore if there does exist a such a number it will be in the form n = 210b + 1(b is a natural number).
Trying 211 = 210(1)+1 which gives a remainder 1 when dividing by 2,3,5,7 ,but 211 is not divisible by 11. Then let b = 2 giving us 421 which gives us the same problem.
If we keep repeating this process we get that 2101 = 210(10) + 1, but 2101 unlike the other numbers is divisible by 11 along with gives the remainder 1 when divided by 2,3,5,7.
Thus 2101 is the solution.
