Hints and solutions to Look for a pattern #5

Two parts today:

Part 1: Fill in the missing numbers in the table with the units (ones) digit of the corresponding power of 2. For example, since 2⁸ = 256, there is a 6 (the ones digit of 256) under 2⁸ in the table. Look for a pattern!

Power of 2	21	22	23	24	25	26	27	28
Units digit	2	4	?	?	?	?	?	6

• More calculations! But you don't have to do the entire multiplication out to get just the last digit. In fact, if you just keep multiplying by 2 and throwing away all but the last digit, things will work out. Or, you can actually do the multiplication all the way out, and look for the pattern (it should repeat every fourth number). The filled in table looks like:

  Power of 2	21	22	23	24	25	26	27	28
  Units digit	2	4	8	6	2	4	8	6

Part 2: Now use your pattern to predict the units digit for these:

212

236

29

225

239

242

• you can see that the last digits will repeat 2,4,8,6,2,4,8,6,... This implies that the units digit for 2¹² and 2³⁶ is 6, foor 2⁹ and for 2²⁵ it's 2 (since the exponents are one more than a multiple of 4), for 2³⁹ the units digit is 8, since 39 is three more than a multiple of 4, and for 2⁴² it's 4.