Name: Kim
Level: 6 - 9, middle
Asking: Student

Question:
What are the last two digits of:

  1. 3 to the power of 1994

  2. 7 to the power of 1994

  3. 3 to the power of 1994 + 7 to the power of 1994

  4. 7 to the power of 1994 - 3 to the power of 1994
Hi Kim,
The key here is to see the pattern.
3 to the power of 1=3
3 to the power of 2=9
3 to the power of 3=27
3 to the power of 4=81
3 to the power of 5=243
3 to the power of 6=729
3 to the power of 7=2187
3 to the power of 8=6561
3 to the power of 9=19683
The last digit repeats in a cycle of length four; 3,9,7,1,3,9,7,1,3...
If you divide 1994 by 4 you find the 1994 = 4 * 498 + 2.

Thus,
3 to the power of 1992 ends in a 1
3 to the power of 1993 ends in a 3
and
3 to the power of 1994 ends in a 9

Cheers,
Penny

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