Question

smallest positive prime factor of the integer 2005^2007+2007^2005​

Answer 1

Answer:

5

Step-by-step explanation:

Answer 2

The smallest positive prime factor of the integer $2005^{2007}+2007^{2005}$ is 2

Explanation:

We know,

$2005=5\times 401\\2007=3\times 3\times 223$

Now,

$2005^{2007}+2007^{2005}\\=(5\times 401)^{2007}+(3\times 3\times 223)^{2005}$

As we can see that they have no common factors, so we may have to add them. But we can solve this question without adding them.

If we look closely,$2005^{2007}$ the one's place is 5 and if we look at the power it's 2007.

We know if any no. has a 5 in its one place and it is raised to any power, then the final no. will also have 5 in its one's place.

If we look closely again, $2007^{2005}$ the one's place is 7

Now, whenever 7 is at one's place, we get a pattern as to what will be the one's place in the final no.

7 raised to 1 has 7 at one's place

7 raised to 2 has 9 at one's place

7 raised to 3 has 3 at one's place

7 raised to 4 has 1 at one's place

7 raised to 5 has 7 at one's place

7 raised to  6 has 9 at one's place

7 raised to 7 has 3 at one's place

7 raised to 8 has 1 at one's place

7 raised to 9 has 7 at one's place

7 raised to 10 has 9 at one's place

We can see that the cycle repeats after every 4 nos.

Now,$2007=501\frac{3}{4}$. This means we have to see the third part of the cycle. The one's place will therefore be 1

If $2005^{2007}$ has 5 at its one's place and  $2007^{2005}$ has 1 at its one's place, their sum will have 6 at one's place. As 6 is divisible by 2, so it is the smallest prime factor.

To learn more about prime factors, click on the links below:

https://brainly.in/question/37691266

https://brainly.in/question/1247261

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