Counting the palindromes

Number Theory Level 3

How many palindromes or palindromic numbers are there between 4×10⁴ and 6×10⁶ ? Here , a palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16461) that remains the same when its digits are reversed.

The answer is 6500.

1 solution

Sathvik Acharya
Jan 16, 2021

$\begin{aligned} 4\times 10^4&<N<6\times 10^6 \\ \implies 40000&<N<6000000 \end{aligned}$ Case 1: $N$ is a $5$ -digit palindrome.

$\;\;\;\;\;\;\;\;\;\;N$ is of the form $\overline{abcba}$ where $4\le a\le 9,\; 0\le b\le 9$ and $0\le c\le 9.$ So the total number of possibilities for $N$ in this case is $6\times 10\times 10=600$ .

Case 2: $N$ is a $6$ -digit palindrome.

$\;\;\;\;\;\;\;\;\;\;N$ is of the form $\overline{abccba}$ where $1\le a\le 9,\; 0\le b\le 9$ and $0\le c\le 9.$ So the total number of possibilities for $N$ in this case is $9\times 10\times 10=900$ .

Case 3: $N$ is a $7$ -digit palindrome.

$\;\;\;\;\;\;\;\;\;\;N$ is of the form $\overline{abcdcba}$ where $1\le a\le 5,\; 0\le b\le 9,\;0\le c\le 9$ and $0\le d\le 9.$ So the total number of possibilities for $N$ in this case is $\\ \;\;\;\;\;\;\;\;\;\; 5\times 10\times 10\times 10=5000$ .

Therefore, there are $600+900+5000=\boxed{6500}$ palindromic numbers between $4\times 10^4$ and $6\times 10^6$ .

Verification

def is_a_palindrome(N):
    return str(N)==str(N)[: :-1]
Palindrome=[N for N in range(4*10**4, 6*10**6)if is_a_palindrome(N)]
print(len(Palindrome))

Sathvik Acharya - 4 months, 3 weeks ago

I solved it in the same way !.!.!.!

Sudipto Podder - 4 months, 3 weeks ago

I think at the end of case 3 you have messed with the term '1000' . It should be 5000. However it's really negligible !.!.!.

Sudipto Podder - 4 months, 3 weeks ago

Thanks, I have corrected it :)

Sathvik Acharya - 4 months, 3 weeks ago

Counting the palindromes

The answer is 6500.

1 solution

0 pending reports