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I have an exam coming up an this will be one style of question can anyone please walk me through how it is done?

What is the least-significant decimal digit of $1002^{3755}$?

MJD
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dunika
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2 Answers2

5

The idea is: You need to calculate mod 10. And since you only need mod 10, you can restrict to $2^{3755}$. And that is easy, because the results of the power of 2 end on 2,4,8,6,2,4,8,6, etc. So you see, every 4th time you find the same value. Therefore the exponent 3755 has the same last digit as the exponent 3 (since 3755 = 3 mod 4) which is 8.

The standard way to calculate $2^{3755}$ would be:

  • Find that $2^5 = 2 mod 10$.
  • Then $2^{3755} = 2^{5*751} = (2^{5})^{751} = (2)^{751} = (2)^{5*150 +1} = (2^5)^{150} *2=(2)^{5*30} *2=2^{30}*2 = 2^6*2=(2)^{5+1}*2 =(2)^5 *2^1*2= 2*2*2 = 8$

So 8 should be the correct answer.

Bernd
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  • I kind of get what your saying would it be possible for you to do out the question so I can see it fully? – dunika Jan 09 '14 at 14:57
  • Where is the *2 at the end coming from? – dunika Jan 09 '14 at 15:16
  • Updated it for the correct numbers - there was a mistake in there. The 2 comes from the +1 in the exponent with 2^(a+b) = 2^a 2^b – Bernd Jan 09 '14 at 15:23
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The "least-significant" digit is the rightmost one. For example, in the numeral 769, the least-significant digit is the 9. It's "least significant" because it only counts for 9 in determining the value of 769. The 6 is actually worth 60; the 7 is worth 700.

So the question is really asking for the units digit of $1002^{3755}$, which is the same as asking for the residue of $1002^{3755}$ mod 10. The first thing to do is to observe that $1002^{3755} \equiv 2^{3755}\pmod{10}$.

Does that help?

MJD
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