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...
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Take a look at this. 41=4 , 42=16, 43=64, 44=256.
In other words, 4odd=...4 and 4even=...6.
So, 4odd+4even=...10
Now we are going to take the terms in pairs.
1+(41+42)+...+(42011+42012)
We know that terms inside the brackets are going to have 0 as the rightmost digit. And adding all of them up together will have 0 as the rightmost digit.
So if we add the 1 that is outside the bracket (the leftmost 1), the last digit's going to be 1.
Hope this helps!
Consider the following:
42k≡4(mod10) and
42k+1≡6(mod10)
Therefore 1+41+42+43...42012≡(mod10)
Observe all the 4′s and 6′s pair up and we get the remainder 1 mod 10. So the rightmost digit is 1
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.
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.
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as 1+4^1+4^2+…4^2012 = 1+4+6+...+6 = 1+0+....0 = 1
Take a look at this. 41=4 , 42=16, 43=64, 44=256. In other words, 4odd=...4 and 4even=...6. So, 4odd+4even=...10 Now we are going to take the terms in pairs. 1+(41+42)+...+(42011+42012) We know that terms inside the brackets are going to have 0 as the rightmost digit. And adding all of them up together will have 0 as the rightmost digit. So if we add the 1 that is outside the bracket (the leftmost 1), the last digit's going to be 1. Hope this helps!
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A slightly cleaner way (which is still saying that same thing) is to say that for n≥0, 42n+1+42n+2=42n(4+16)=42n×20, hence has a units digit of 0.
This avoids the slight issue of 40=1, while claiming that 4even=…6.
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Thank you for pointing it out.
Consider the following: 42k≡4(mod10) and 42k+1≡6(mod10) Therefore 1+41+42+43...42012≡(mod10) Observe all the 4′s and 6′s pair up and we get the remainder 1 mod 10. So the rightmost digit is 1
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Latex error: It should be 1+41+42+43...42012≡1+4+6+4+6.....(mod10)
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right
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.
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.