Divisiblity Tests

I describe a method to generate a divisibility test for the divisor 13.

For a dividend B, express B in terms of its digits, i.e,
B = 10X + Y , where Y is the ones digit and X are the digits left of Y.

Working in modulo 13, if B is divisible by 13, then
10X + Y ≡ 0 .

We then propose a divisibility test that uses X and Y to check the divisibility. We propose
X - KY ≡ 0 ,
meaning that we subtract K times Y from the X digits and test whether 13 divides it.

Manipulating and substituting,
10KY + Y ≡ 0 => (10K+1) ≡ 0 .

Solving the equation, we obtain a value of K = -4. Hence, to test whether a number is divisible by 13, we take the ones digit, multiply it by 4 and add it to the digits on the left. If the result is divisible by 13, then the number is divisible.

To demonstrate the use of the test, let us test the numbers 1234, 2468, and 1781. For 1234, 123 + 4*4 = 139, which is clearly not divisible by 13. Hence, 1234 is not divisible by 13. For 2468, 246 + 4*8 = 278. We can recur the test, 27 + 4*8 = 59, and hence 2468 is not divisible by 13. Lastly, for 1781, 178+4 = 182, 18 + 4*2 = 26. Hence, 1781 is divisible by 13.

The method for generating the divisibility test is general, and can be extended to other numbers. However, for larger numbers, it might be necessary to compute the tens and hundreds digits (and increasingly higher powers of ten), so it might not be feasible for very large numbers.

As a last note, divisibility tests are useful for trivial tasks such as prime factorization using your head.

Musings on Transport Subsidies

During a dialogue session with Macpherson residents on Sunday, Transport Minister Raymond Lim, in response to questions on the rising cost of public transport, revealed that it would take a further 1.5 percentage point hike in GST if bus and train rides were to be made completely free.

On certain levels, the argument is attractive. As a libertarian and a minarchist, one of my beliefs is that governments should be as small and limited as possible. The argument for minarchism is similar to that raised by the Transport Minister- that ultimately, governmental interference (for example, in the form of subsidies) is circular, taking with one hand and giving with the other. More importantly, the objection is that such interference is pointless and inefficent.

Though the arguments for minarchism are actually deeper, the theory does appear to stick to the issue of public transport. It is a simple solution to demand transport subsidies from the Government, but ultimately, is not the Government funded by the taxpayer? In effect, transport subsidies are tantamount to the public transport user being subsidized by the private transport user, which may or may not be desirable, depending on your views on wealth redistribution.

There is, however, one mistake in the prior reasoning. That mistake is to assume that the game is zero-sum, and that subsidies take X from a person and distributes X to another. and that there is no change in overall utility. For certain goods, especially public goods, this may not be true. In some cases, total utility can be increased by redistribution. One notable example is that of economic stimulus packages, which, by taking tax money and redistributing it to the citizens, might initially appear to be foolish, but is actually somewhat a prudent measure, since it encourages economic growth in the hope that the tax money lost could be more than recouped by the growth in the tax base.

Having raised the idea that subsidies might not be zero-sum, let us try to explore how this can be so. To speed up affairs, let us ask whether a sum X spent in transport subsidies can possibly generate a greater sum Y in additional economic growth. To answer the question, we need to study the effects of the additional mobility granted by the corresponding reduction in public transport costs.

Firstly, I could postulate that the gain in mobility may generate a weak rise in consumer spending, since we are less deterred from going out, though I admit that any rise in spending would most likely be minimal. A second effect is that retail competitiveness would increase, as consumers are less likely to be locked in to vendors from the neighborhood. This might lead to smaller shops (pop-and-mom outfits) being negatively affected, since the advantage of locale would be eroded.

A more important effect is that of jobs. Traditionally, distance (from the home to the workplace) is one barrier to the choice of jobs available to a worker. While there may be many job opportunities, many jobs are not possible or at least less attractive due to the commuting distance, and also commuting cost. Corresponding reductions in the cost of commuting may see a rise in employment rates, or of worker quality.

It might be possible that subsidies for public transport may generate more utility than their cost, though very careful studies must be made to quantify arguments for it, as I admit that the arguments I put forward in this post are at this stage entirely hypothetical. In particular, the issue of how the subsidies are to be funded should also be taken into account, as certain forms of funding are likely to destroy or erode gains made from the subsidies. For example, additional consumption taxes are likely to dampen or even reverse growth in consumer spending. Hence, more research would be desirable to shed more light onto this matter.

How To Procrastinate

I have acquired some absurdly bad procrastination habits lately. Lots of things are being put off, though these are not by any means important matters, which is both assuring (well, it's only the unimportant things) and terrible (well, even the unimportant things are being put off!).

One telling indicator: My Firefox starts up with many tabs, and I think the leftmost two or three tabs have been there for a couple of days. My "TEMP" bookmark folder is increasing in size too.

I've got to devote more time to actually reading those webpages.

The Amazing Tale of Intrigue

The wicked witch had run out of her supply of children and was feeling hungry. Due to the poor economic climate, buying children off the market was costly, and so the witch decided to do things the old way, which was to catch some.

After breaking off two large baskets of sweets from her candy house, the witch moved to the traditional kid-luring location. The efficacy of the bait amazed even herself, as a child lunged towards her before she even reached the location.

But it was not to be! The child was not prey, but predator! The brave blue boy stunned the witch with a leap attack, and carried off the two baskets of lures. The unprecedented attack shocked the witch - could this boy be the fabled witch-slaying hero? The witch recalled the many cases of mysterious witch-deaths, such as one dying after "accidentally" climbing into the kitchen oven, and another being crushed to death by a falling house.

No chances could be taken, the crone thought. Cackling, she took out a spell scroll, disguised as currency, from her purse. She started chanting the activation words, but the child-hero acted swiftly, using kungfu to interrupt the spell. He focused his internal energies into his palms and blasted the witch. The power of the spell and the energy blast mixed together, and neutralized each other.

Her spell neutralized, the witch had to use the last resort, which was also kungfu. She raised her palm to attack, but her actions were far too slow. The child-hero had the time to change his clothes, and to take out a hose and spray water over the witch. The witch writhed in pain, as all witches were weak to water. One notable relative living in the west was even slain by immersion.

The child-hero took the witch back to the secret headquarters for interrogation. This move would prove to be a mistake, since there was a secret double agent in the organization. The boy was caught unawares by a sneak attack and was defeated.

The witch took the boy back to her candy home, where she intended to cook the boy. Unexpectedly, a deus ex machina happened, and a taxi popped out from empty space. Riding inside it was Captain Mustache, one of the most famous superheroes in the world. The greatest battle of the century was about to begin...

The Game

There is a game which everyone plays. It is a queer game, though. Some are unaware that they are playing. Others play without knowing their final goals in the game. Still, all play the game.

There are many possible objectives in the game. Some choose to pursue material wealth, seeking to collect the most riches. Others choose to collect the most company, amassing the most friends and favor. Yet some more play the game seeking a simpler victory, one that is completed with a partner. There are indeed many ways to win the game.

Curiously, while the game is not difficult to win, many dawdle aimlessly. They pursue meaningless goals which are unrelated to their winning conditions, hence wasting time and resources. Some pursue grades blindly with their final aim being wealth and not education, and yet these players are unaware of the other game plans that would bring them closer to their goal, and faster. Other players pursue wealth and career, but are unaware that their winning condition is happiness, and that their futile moves bring them further and further away from winning.

It seems trivial, but the first step to winning the game is to learn what the winning condition is. Otherwise, one would find himself playing the wrong game, and losing.

Are you playing the right game?