Question

Find the smallest prime number

dividing the sum ( 3^11 + 5^13 ).

explain step by step.

Answer 1

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.

Answer 2

Let's observe something before going to the actual question.

1) 3+5 = 8, 9+7=16

Sum of two odd number is always a even number.

2) 3^1= 3,

3^2=9

3^3=27

so, any power of 3 is ending in odd number,hence 3^n is always odd number.

3) 5^1 = 5

5^2= 25

5^3=125

5^4=625


We observe that any power of 5 is ending with 5, which is hence a odd number.

4) 456/2 gives remainder 0, it is a even number. So all even numbers are divisible by 2 as even numbers end with 0,2,4,6,8.

Hence, 3^11 is odd number

5^13 is odd number.

So, 3^11 + 5^13 is the sum of two odd numbers. We know that sum of two odd numbers is even number.

As all even numbers are divisible by 2, the sum of 3^11+5^13 is divisible by 2.

We also know that there is no least prime number than 2.

Hence, Least prime number dividing the sum of 3^11+5^13 is 2.

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hope helped!