# Last Digit of N

N = 11986 + 21986 + 31986 + 41986.

What is the last digit of N?

Answer

To find the last digit on N we have to find the last digit of each of the terms adding up to N: 11986, 21986, 31986, and 41986. Of course, one could always compute the result one power at a time. Steady and slooowly. The world record for the fastest time to complete the official 250-piece jigsaw puzzle of the Guinness World Records is 13 minutes and seven seconds. Mathematics shouldn’t necessarily be a race against time, but it should be about finding the answer in the most efficient, i.e. interesting, way possible.

Let’s start with 21986.

Can we write it as product of factors whose last digits are easy to determine?

What number, power of 2, yields the same last digit when raised to any power?

It’s 16 = 24. 16 raised to any power will give a number that ends in 6 (because 6 raised to any power gives a number that ends is 6. Always.)

You can rearrange 21986 as:

21986 = 24×496+2 = 24×496 x 22 = (24)496 x 4 = 16496 x 4

16496 ends in 6, and 6 x 4 = 24, so the last digit of 21986 is 4.

How about 41986?

It’s very similar to 21986.  Try it on your own, and then check your answer below.

41986 = 42×993 = 16993.

Last digit of  41986 is 6.

How about 31986?

Try it on your own, and then check your answer below.

31986 = 34×496+2 = (34)496 x 32 = 81496 x 9.

Since 81496 ends in 1, then last digit of 31986 is 1 x 9 = 9.

Last digit of N is last digit of 1 + 4 + 9 + 6 = 20. It’s just … 0.

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