Category Archives: Index Laws

September 17, 2025 · 7:03 am

Divisibility Rule for 11

I was working on a question and involved 11 and I wondered what the divisibility rule was?

So then I had a bit of a think about it.

Let N be a number divisible by 11. The N (mod11)=0

    \begin{equation*}N=a_n\times10^n+a_{n-1}\times10^{n-1}+a_{n-2}\times10^{n-2}+...+a_0\times10^0\end{equation}

    \begin{equation*}0=a_n\times10^n (mod11)+a_{n-1}\times10^{n-1}(mod11)+a_{n-2}\times10^{n-2}(mod11)+...+a_0\times10^0(mod11)\end{equation}

Now 10 (mod11)=10 which is congruent to -1 because 10-(-1)=11, which is a multiple of 11.

Thus

    \begin{equation*}0=a_n(-1)^n+a_{n-1}(-1)^{n-1}+a_{n-2}(-1)^{n-2}+...+a_0(-1)^0\end{equation}

Odd powers will be negative and even positive.

So if we start at one end of the number and add every second digit (i.e. first digit plus third digit plus fifth digit etc.) and then subtract the other digits (i.e. second digit, fourth digit, six digit, etc.), if that equals zero then the number is divisible by 11.

For example, is 1756238 divisible by 11?

1+5+2+8-7-6-3=16-16=0

Hence 1756238 is divisible by 11

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November 29, 2024 · 8:11 am

Interesting Equation

I think this one is doing the rounds, I first saw it here.

    \begin{equation*}2^x3^{x^2}=6\end{equation}

x=1 is the obvious answer, 2^1\times 3^1=6, but are there more answers?

This was my approach

    \begin{equation*}ln(2^x3^{x^2})=ln(6)\end{equation}

    \begin{equation*}ln(2^x)+ln(3^{x^2})=ln(6)\end{equation}

    \begin{equation*}xln(2)+x^2ln(3)-ln(6)=0\end{equation}

    \begin{equation*}ln(3)x^2+ln(2)x-ln(6)=0\end{equation}

A quadratic equation.

Hence,

    \begin{equation*}x=\frac{-ln(2)\pm\sqrt{(ln(2))^2-4(ln(3))(ln(6))}}{2ln(3)}\end{equation}

I then used my calculator

Hence x=1 0r x=-1.631

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