Question

Find the smallest prime number dividing the sum 3^13 +5^13​

Answer 1

Any product of odd numbers is always odd.
The sum of two odd numbers is always even.
Since, 3`11 and 5`13 are both the product of odd numbers, so they are both odd
So 3`11 + 5`13 is the sum of two odd numbers and is therefore even.
Any even number is divisible by 2. Therefore 2 is the smallest prime that divides 3`11 + 5`13
Also, 3`11 + 5`13 = 1220880272
Therefore, the smallest prime number dividing the sum 3¹¹ + 5¹³ is 2.

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Answer 2

Step-by-step explanation:

let the smallest prime no. be x =2

3 = 4-1 = 2x-1

5 =4+1= 2x+1

so 3^13+5^13 can be written as

f(x)=(2x-1)¹³ + (2x+1)¹³

dividing with 2 => dividing f(x) with x then

remainder =f(0)

f(0)=(-1)¹³+1¹³=-1+1=0

=> f(x) is divisible by x as remainder =0

hence x is a factor of f(x)

=> 2 is a factor of 3¹³+5¹³

hence 2 is the smallest prime number that divides the sum

3¹³+5¹³