# Browsing category number theory

## Primes or not?

I’m in the process of clearing out old bookmarks, and stumbled on this puzzle from @jase_jwanner: Prime or not prime? No calculators allowed!a. 23567897614^2 - 1b. 34564344^3 -1c. 76543556556625731d. 345643554^{10} - 169 — Jase (@jase_jwanner) August 27, 2016 I shall give you a moment to ponder these, and put my

## Nonupling

My cunning plan, back last August, was sadly foiled: @christianp refused to rise to the bait. I'd written a post about finding the smallest number such that moving its final digit to the front of the number doubles its value. It turned out, to my surprise, to be 17 digits

## Several Strings of 1s

This puzzle was in February's MathsJam Shout, contributed by the Antwerp MathsJam. Visit mathsjam.com to find your nearest event! Consider the set ${1, 11, 111, ...}$ with 2017 elements. Show that at least one of the elements is a multiple of 2017. The Shout describes this one as tough; you

## On the trick with 9s

Pick a number, any number1 Let's say 24,876,028, one of my favourites in the 20-millions. I can tell by looking at it that it's one more than a multiple of 9. It's not that clever a trick: I just added up the digits (getting 37), and repeated the trick with

## Ask Uncle Colin: The Last Two Digits

Dear Uncle Colin, I've been asked to find the last two digits of $19^{100}$. For what reason, I cannot tell. However, my calculator bums out before I get to $19^{10}$! -- Many Other Digits, Unfindable Last Ones Hi, there, MODULO! What do you know, the clue to your problem is

## Ask Uncle Colin: A Modulo Mistake

Dear Uncle Colin, I wanted to work out $3^{41}\mod 13$: Wolfram|Alpha says it's 9, but MATLAB says it's 8. They can't both be right! What gives? MATLAB Obviously Doesn't Understand Logical Operations Hi, MODULO! First up, when computers disagree, the best thing to do is check by hand. Luckily, you

## Why is $(2+\sqrt{3})^n$ nearly an integer?

Funny thing. Type $(2 + \sqrt{3})^{20}$ into Wolfram Alpha. (Or, if you're really lazy, click this link.). It's 274,758,382,273.999999999996 or so. The higher the power you pick, the closer $(2 + \sqrt{3})^n$ gets to an integer value -- although it never quite gets there, because $\sqrt{3}$ is irrational. So, how

## A neat number trick: digital roots and modulo-9 arithmetic

This post is inspired by a question asked by Dan - thank you, Dan! So here's the gist of Dan's question: Take a random sum, e.g. 496866 + 446221 = 943087. Add up all the digits in each number (39, 19 and 31). Keep adding up the digits in each