Challenge Question

[sanjivsp]

thought it was interesting that difference in the tens unit is always 8. any reason?

036
x16
x96
x76

Question
The tens digit of 6 to the 17th power is
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9
Answer

We know that there must be a pattern, since we can’t be expected to expand 617 out to all its digits. In other words, we must be able to spot a repeating cycle of digits.

The only way forward is to compute tens digits for powers of 6, starting with 61, and see what we get. To go up, multiply the previous result by 6 and drop any higher digits than the tens, but we have to keep the units digit (which, as we’ll see, will be 6 every time).

61 = 6 (no tens digit)
62 = 6 × 61 = 36 (tens digit = 3)
63 = 6 × 62 = ..16 (tens digit = 1)
64 = 6 × 63 = ..96 (tens digit = 9)
65 = 6 × 64 = ..76 (tens digit = 7)
66 = 6 × 65 = ..56 (tens digit = 5)
67 = 6 × 66 = ..36 (tens digit = 3)

[Ben Ku]

Please cite the source (author) of this problem. We cannot reply unless a source is cited (and, if no source is cited, we will have to delete the post!). Thanks.