#prove divisibility by induction

1 messages · Page 1 of 1 (latest)

delicate oysterBOT
#
  1. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
  2. Wait patiently for a helper to come along.
  3. Once someone helps you, say thank you and close the thread with:
    +close
  4. Feel free to nominate the person for helper of the week in #helper-nominations
  5. Do not ping the mods, unless someone is breaking the rules.
  6. If you're happy with the help you got here, and the server overall, you can contribute financially as well:
thorny quartz
#

First of all, it's probably worth setting n = 2k + 1. Then we need to prove that 2^(6k + 4) + 2*6^(2k + 1) is divisible by 28 for all nonnegative integer k.
First, it's probably worth simplifying the expression a bit more. Try factoring out some stuff.

thorny quartz
#

Hm... We've transformed the problem to divisibility of 4*64^k + 3*36^k by 7.
So, we check k = 0, it works, all is good.
I am also having trouble with the step, though.

#

Nah, it probably is. I'm just not clever enough to notice how.

gritty walrus
#

Try using that 64 = 7*9+1 and that 36 = 7*5+1

modern pewter
#

In the same vein of @gritty walrus's suggestion, you can replace the following in your last line:
2^4 = 16 = 7 * 2 + 2
and
3^2 = 9 = 7 * 1 + 2

modern pewter
#

Replace "any number" by "any integer" and then it looks good to me