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In the past few days, I, my friends, and a whole lot of Twitter people have been trying to beat each other’s scores in the game “Is This Prime?”.
The game itself is simple; you are shown random integers on the screen and you need to guess whether they are prime or composite. Since most presented numbers are between 1 and 200, after a couple of games, players naturally memorize them. However, this is a good opportunity for students to review some main number division rules.
- Numbers that end with an even digit are divisible by 2. If the number formed by the last 2 digits of a number is divisible by 4, the original number is also divisible by 4. If the number formed by the last 3 digits of a number is divisible by 8, the original number is also divisible by 8.
- 536 is divisible by 2 because 6 is an even digit
- 1348 is divisible by 4 because 48 is divisible by 4
- 71824 is divisible by 8 because 824 is divisible by 8
- Numbers that end with 5 are divisible by 5. Numbers that end with 25, 50, 75, or 00 are divisible by 25.
- 45 is divisible by 5
- 675 is divisible by 25
- Numbers whose sum of digits is divisible by 3 are divisible by 3. Numbers whose sum of digits is divisible by 9 are divisible by 9.
- 144 is divisible by 3 because 1+4+4=9 is divisible by 3
- 1638 is divisible by 9 because 1+6+3+8=18 is divisible by 9
- If the difference between the sum of the digits in odd places and the sum of the digits in even places is divisible by 11, the number is divisible by 11.
- 121 is divisible by 11 because 1+1-2=0 is divisible by 11
- 209 is divisible by 11 because 2+9-0=11 is divisible by 11
- 1628 is divisible by 11 because 1+2-6-8=-11 is divisible by 11
- If the number before the last digit minus twice the last digit is divisible by 7, the original number is also divisible by 7.
- 161 is divisible by 7 because 16-2×1=14 is divisible by 7
- 371 is divisible by 7 because 37-2×1=35 is divisible by 7
- 1589 is divisible by 7 because 158-2×9=140 is divisible by 7
All the rules above apply in both directions, e.g. if the sum of the digits of a number is not divisible by 9, then the number itself is also not divisible by 9. There are more complicated rules that apply to larger numbers but the chances are you will never get to use them. If you are curious to learn more about them, go to the bottom of this article.
Once we know the main number division rules well, we are ready to play the game! Here are a few tips for getting high scores:
- Memorize as many numbers as possible. Knowing the multiplication table up to 10×10, it should be easy to learn by heart whether each number up to 100 is prime or composite.
- Pay attention to the last digit. If it is 5, then the number is composite (unless it is =5).
- Check whether the sum of the digits is divisible by 3. If it is, then the number is composite (unless it is =3).
- If the number is between 100 and 300, check whether the sum of the first and the third digits equals the second digit. If this is true, then the number is divisible by 11, and therefore it is composite. 209 is the only other number in this range divisible by 11.
Good luck playing and let us know if you beat our personal record of 67 points!
Primes between 1 and 300:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199
211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293
51, 57, 87, 91
119, 133, 161, 169
209, 217, 221, 247, 253, 259, 287, 289, 299
Some of you may be wondering why the rules listed above work and whether we can create similar rules for larger numbers. Here are some explanations for the curious among you:
Rule for division by 3
Consider the 5-digit number ABCDE. It can be written as:
ABCDE = 10000A + 1000B + 100C + 10D + E
Since 10000 = 3 × 3333 + 1, 1000 = 3 × 333 + 1, 100 = 3 × 33 + 1, and 10 = 3 × 3 + 1, we can see that:
ABCDE = 3 × (3333A + 333B + 33C + 3D) + (A + B + C + D + E)
Therefore, ABCDE is divisible by 3 if and only if (A + B + C + D + E) is divisible by 3.
Rule for division by 11
Consider again the number ABCDE. Since 10000 = 11 × 909 + 1, 1000 = 11 × 91 – 1, 100 = 9 × 1 + 1, and 10 = 11 – 1, we can see that:
ABCDE = 11 × (909A + 91B + 9C + D) + (A – B + C – D + E)
Therefore, ABCDE is divisible by 11 if and only if (A – B + C – D + E) is divisible by 11.
Rule for division by 7
Once again, consider the number ABCDE. Notice that it can be written as:
ABCDE = 10 × ABCD + E
Now, let us find a number X such that 10X gives remainder 1 when divided by 7. Such number is X = 5. Indeed, 5 × 10 = 50 = 7 × 7 + 1. Therefore, the following statements are equivalent:
- ABCDE = 10 × ABCD + E is divisible by 7
- 5 × ABCDE = 49 × ABCD + ABCD + 5E is divisible by 7
- ABCD – 2E is divisible by 7
Rules for division by 13, 17, 19, etc.
The idea of the rule for division by 7 can be applied to rules for divisions by higher numbers. For example, here is how we can find a rule for division by 13:
- Find the smallest positive integer X, so that 10X is divisible by 13. Such number is X = 4. Then, ABCDE is divisible by 13 if and only if 4 × ABCDE is divisible by 13.
- Rewrite 4 × ABCDE as:
4 × ABCDE = 39 × ABCD + ABCD + 4E
- Conclude that ABCDE is divisible by 13 if and only if ABCD + 4E is divisible by 13.
As an exercise, try to deduce a similar rule for division by 17!
For deeper understanding of how division of integers works, we recommend our more enthusiastic readers to look into Modular Arithmetic.
The Puzzle TOAD is a website, created by four Carnegie Melon professors (Tom Bohman, PO Shen-Loh, Alan Frieze, Danny Sleator), where you can find a growing collection of ingenious math brain teasers. Unlike Puzzle Prime, The Puzzle Toad is targeted exclusively towards math and computer science majors. Students who are preparing for college Olympiads will find the problems particularly useful. Check out The Puzzle TOAD by clicking the banner below.
Minecraft: Magnetic Travel Puzzle (M:MTP for short) is a travel game by ThinkFun in which the goal is to arrange 3 types of objects, each coming in 3 different colors, in a 3 by 3 grid, such that certain conditions are satisfied.
As you progress through the 40 included challenges, the types of conditions you encounter become gradually more complex. While in the beginning you may be given all the colors of the objects with one clue and all the types of the objects with another, later on you need to analyze 5 or 6 clues at once, which makes the game more challenging and fun. That being said, at the hardest levels, M:MTP is still relatively easy, so experienced puzzlers will probably breeze through it within an hour or two.
At its core, M:MTP is identical to ThinkFun’s previously released Clue Master. Both games are presented in the form of magnetic notebooks, so they are easy to pick up and travel around with. The illustrations of the Minecraft edition are all based on the popular video game, so its fans may be particularly appreciative.
If you are looking for a casual puzzle to pass an hour or two on a road trip, then M:MTP would be a great choice. I only wish there were more challenges included, especially more difficult ones.
- 1 player, 8 years and up
- 40 challenges with increasing difficulties
- easy to transport and play on the go
- cool Minecraft based art
- most puzzles can be solved with a few simple techniques
“The Boat” is a graphic novel adaptation of Nam Le’s book, presented as an immersive webpage experience by Matt Huynh. With beautiful artwork and stunning effects, the novel tells the story of a 16-year old Vietnamese refugee, embarking on a dangerous trip across the sea. Click the banner below and scroll your way through this captivating, moving, and harrowing tale.