Solve without using anything except brain!!

Algebra Level 4

$21870000901$ is

Please don't use any code or internet or calculator.

Composite Prime neither all Perfect square Perfect cube

3 solutions

Aryan Sanghi
May 24, 2018

$21870000901 = 30^7 + 30^2 + 1$

Let $x = 30$

$30^7 + 30^2 + 1=x^7 + x^2 + 1$

$= x^7 - x + x^2 + x + 1$

$= x(x^6-1) + x^2 + x + 1$

$= x(x^3 + 1)(x^3 - 1) + x^2 + x + 1$

$= x(x^3 + 1)(x - 1)(x^2 + x + 1) + x^2 + x + 1$

$= (x^2 + x + 1)(x(x^3 + 1)(x-1)+1)$

We can see number has more than 2 factors. Hence number is $\color{#3D99F6}{\boxed{\text{composite}}}$

What others say that dividing by 7 is what I did. And it seems reasonable to think that as it is difficult to figure out that it's of the form you wrote . Isn't it?

Md Zuhair - 2 years, 3 months ago

Yes, you are right. But, when I posted the question, I didn't knew that it is divisible by 7. And, this method what I wrote is useful in RMO Bhaiya. :)

Aryan Sanghi - 2 years, 3 months ago

Yes obviously its useful for written exams like RMO, ISI and CMI. If they give it... Its a tough game to crack the question... you dont know na... if $x^7$ will be given or $x^9$ . As you set the question, Its easy for you to think.. Isnt it? :)

Md Zuhair - 2 years, 3 months ago

Yes, you are right.

Aryan Sanghi - 2 years, 3 months ago

X X
May 19, 2018

$21870000901=7^2\times19\times197\times119243$

Did you search in internet sir??

Aryan Sanghi - 3 years ago

I divided by 7,and get the answer.But yes,I did use a code to do the rest of the things.

X X - 3 years ago

Nick Turtle
May 19, 2018

Divide the number by $7$ (mentally). You will find that it is indeed divisible. Thus, the number is composite.

Solve without using anything except brain!!

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