That number is so selfish

Number Theory Level 3

Find the smallest 3 digit narcissistic number that has a tens digit of 5.

This one is meant to be done by hand given the following hint:

This requires very little work if you guess and check in a smart way

Details:

A narcissistic number is a number that when its digits are individually raised to the n-th power and summed, they equal the original number.

Example:

$407=4^3+0^3+7^3$

The answer is 153.

2 solutions

Satvik Choudhary
Apr 3, 2015

Let the number be x5y so that x^3+5^3+y^3=100x+5×10+y. Since it is smallest no. I guessed x to be 1 this gives 1+125+y^3=100+50+y. y^3-y=24 By hit and trial y=3 So the number is 153.

Trevor Arashiro
Apr 3, 2015

Begin with

$100a+10b+c=a^n+b^n+c^n$

Since our ten's digit is a 5, $b=5$

We are trying to find the smallest possible, so start by guessing the smallest number: 1

$100(1)+50+c=1+5^n+c^n$

$149+c=5^n+c^n$

Obviously, $n=1,2$ have no solutions.

Checking $n=3$

$c^3-c-24=0$

$(c^3-27)-(c-3)=0$

$(c-3)(c^2+3c+8)=0$

Thus $c=3$ and our smallest number is $153$

Actually, a $n-$ digit narcissistic number $\overline{a_1a_2\cdots a_n}$ satisfies:

$\overline{a_1a_2\cdots a_n}=\sum_{k=1}^n (a_k)^n$

So, you don't need to check with different $n$ 's. For a $3-$ digit narcissistic number, you cannot have any $n$ other than $3$ . If you have any other $n$ , then it is a perfect digital invariant but not a narcissistic number. You can read more about it here .

Prasun Biswas - 6 years, 2 months ago

That number is so selfish

The answer is 153.

2 solutions

0 pending reports