Name That Divisor

Number Theory Level 2

$\large {4^{61}}+{4^{62}}+{4^{63}}+{4^{64}}+{4^{65}}$

The number above is divisible by which of the following?

5 17 3 11 None of the above

1 solution

Rohit Udaiwal
Jan 25, 2016

$\huge {\color{#D61F06}{4^{61}}+\color{#3D99F6}{4^{62}}+\color{#20A900}{4^{63}}+\color{magenta}{4^{64}}+\color{#EC7300}{4^{65}} \\ =\color{#D61F06}{4^{61}}(1+4+4^2+4^3+4^4) \\ =\color{#D61F06}{4^{61}}\cdot 341}$

Now since 341 is divisible by 11,the required number is also divisible by 11.

Note :341 is divisible by 11 because sum of digits at odd place and the sum of digits at even place have a difference of either 0 or a multiple of 11.That is, $\color{#624F41}{3+1}-\color{#E81990}{4}=0$ .

Note the difference could also be negative try $281611$

William Isoroku - 5 years, 4 months ago

Negative numbers can be multiples of 11 too :)

Arulx Z - 5 years, 4 months ago

Name That Divisor

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