Question

What is the greatest integer among the following by which the number $5^5+7^5$ is divisible?

• A. $6$
• B. $8$
• C. $11$
• D. $12$

Answer 1

Answer: (D)

$12$

Now $5^5+7^5=(6-1{)}^5+(6+1{)}^5$

$T_{r+1}{=}^n{C}_r{a}^{n-r}{b}^r$

$5^5+7^5=(6-1{)}^5+(6+1{)}^5=2{[}^5{C}_0\left(6{)}^5\right(1{)}^0{+}^5{C}_2\left(6{)}^3{+}^5{C}_4\right(6\left)\right]$

$=12[6^4+10.6^3+5]$

Greatest Integer $=12$

For $r=1,3,5$ all the terms will cancel out each other.