MAYBE a familiar problem

I have several problems. One of them is: Find is the rightmost digit of 1+4^1+4^2+…4^2012 As it's my 1st discussion I started by picking only 1 problem...

Note by Sheikh Asif Imran Shouborno
8 years ago

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Comments

1

as 1+4^1+4^2+…4^2012 = 1+4+6+...+6 = 1+0+....0 = 1

Anoopam Mishra - 8 years ago

Take a look at this. 41=44^1=4 , 42=164^2=16, 43=644^3=64, 44=2564^4=256. In other words, 4odd=...44^{odd}= ...4 and 4even=...64^{even}= ...6. So, 4odd+4even=...104^{odd}+4^{even}=...10 Now we are going to take the terms in pairs. 1+(41+42)+...+(42011+42012)1+(4^1+4^2)+...+(4^{2011}+4^{2012}) We know that terms inside the brackets are going to have 00 as the rightmost digit. And adding all of them up together will have 00 as the rightmost digit. So if we add the 11 that is outside the bracket (the leftmost 11), the last digit's going to be 11. Hope this helps!

Mursalin Habib - 8 years ago

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A slightly cleaner way (which is still saying that same thing) is to say that for n0n\geq 0, 42n+1+42n+2=42n(4+16)=42n×20 4^{2n+1} + 4^{2n+2} = 4^{2n} ( 4 + 16) = 4^{2n} \times 20 , hence has a units digit of 0.

This avoids the slight issue of 40=1 4^0 =1 , while claiming that 4even=6 4^{even} = \ldots 6 .

Calvin Lin Staff - 8 years ago

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Thank you for pointing it out.

Mursalin Habib - 8 years ago

Consider the following: 42k4(mod10)4^{2k}\equiv 4{\pmod{10}} and 42k+16(mod10)\\4^{2k+1}\equiv 6{\pmod{10}} Therefore 1+41+42+43...42012(mod10) 1+4^1+4^2+4^3...4^2012 \equiv \pmod{10} Observe all the 4s4's and 6s6's pair up and we get the remainder 11 mod 10. So the rightmost digit is 11

Vikram Waradpande - 8 years ago

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Latex error: It should be 1+41+42+43...420121+4+6+4+6.....(mod10)1+4^1+4^2+4^3...4^{2012}≡ 1+4+6+4+6.....\pmod{10}

Vikram Waradpande - 8 years ago

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right

superman son - 8 years ago

Ignoring 1, if we start with 4^1, we will be able to recognise that 4^(odd number) has the last digit as 4, for eg:-4^3 = 64 and 4^5=1024. In the same way, 4^(even number) has the last digit as 6. The sum is congruent to 0(mod10). Now adding 1 in the sequence, we will get the sum congruent to 1(mod10). So 1 is the last digit.

Siddharth Kumar - 8 years ago

Another method to solve this problem, is to observe that the numbers are in geometric progression. So use the sum formula of geometric progression and try to find the last digit either by simple calculation, or modular arithmetic.

Siddharth Kumar - 8 years ago
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