Discrete math 19 Show that there exists an integer multiple

Discrete math

19. ) Show that there exists an integer multiple of 17 that consists entirely of 2\'s and 0\'s.

B) Find one such integer.

Hint: 22,222,222,222,222,222 when divided by 17 leaves a remainder of 2.

Solution

A) If we consider the integer 22,222,222,222,222,220 and divide it by 17, we find that it goes 1307189542483660 times and it leaves a remainder zero.

Or 22,222,222,222,222,220 = 1307189542483660 x 17 + 0

that means this number is completely divisible by 17. That proves that there exists an integer multiple of 17 that consists entirely of 2`s and 01s.

B) So 22,222,222,222,222,220 is one such integer.