What is the Remainder?

What is the reminder of 9991000 divided by 5?

Answer

We first have to determine the possible units digits of powers of 9.

91 = 9

92 = 81

93 = 729

94 = 6561

95 = 59049

We notice that the units digits of powers of 9 alternate between 1 and 9. For even powers of 9, the units digit is 1. For odd powers of 9, the units digit is 9.

Therefore the last digit of 9991000 is 1.

Why do we only care about the last digit? Natural numbers that end in the same last digit have the same remainder when divided by 5.

A number that ends in 1, no matter what its other digits are, will have a remainder of 1 when divided by 5.

The reminder of 9991000 divided by 5 is just 1.

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