Political Calculations
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December 13, 2019

Can you quickly tell if a number is divisible by any of the numbers from 1 to 12 without actually doing the division?

Let's say you have the number 1,512 and it be really helpful if you could determine if it could be evenly divided by any of the numbers from 1 to 12. Could you do it without launching the calculator app on your mobile phone and performing the divisions?

You can, but you'll need to apply the divisibility rules you might have learned a long time ago in school. In case you don't remember them, here they are, where we've included one or two you may never have seen before. Try them with 1,512 and see which apply....

Photo by Gayatri Malhotra on Unsplash - https://unsplash.com/photos/4wF66_KWJxA
  1. This is the easiest divisibility test of all, because all whole numbers are divisible by 1.
  2. Look at the last digit of your number of interest. If it is even (equal to 0, 2, 4, 6, or 8), then your number of interest is divisible by 2.
  3. Add up all the individual digits of your number of interest. If the sum is divisible by 3, so is your number of interest. And if you can't tell right away, repeat this process with the digits of your sum!
  4. Look at the last two digits of your number of interest. If those two digits are a multiple of 4, your whole number will be evenly divided by 4.
  5. Look at the last digit of the number of interest. If that digit is either 0 or 5, your number will be divisible by 5.
  6. You'll need to perform a two-part test to determine if your number is divisible by 6. Specifically, you'll need to perform the divisibility tests for both 2 and 3 on it, where if it passes both tests, then it may be evenly divided by 6.
  7. This is the hardest of the divisibility tests. Split off the right-most digit of your number of interest from the rest of it, and multiply it by 5. Then add the result to the remaining left-side of your number. If the result is a multiple of 7, your original number will be evenly divisible by 7. If you can't tell right away, repeat this process with your result until you can.
  8. Telling if your number can be evenly divided by 8 requires a two-part test. First, do the divisibility test for 4, where if it passes, identify the factor by which you have to multiply by 4 to get the last two digits of your number. Then look at the digit in the hundreds place (the third digit from the right) of your number of interest. If both of these values are even, or if both of these values are odd, then your number will be wholly divisible by 8.
  9. Nine is a multiple of 3, so the divisibility test is a lot like that one. Add up all the individual digits of your number of interest and if the sum is divisible by 9, so is your number of interest. If you can't tell right away, repeat this process with the digits of your sum until you can.
  10. This may be the second easiest of the divisibility tests. Look at the last digit of your number. If it is 0, then your number will be evenly divided by 10.
  11. This is a fun one. Starting with the left-most digit of your number of interest, alternate subtracting and adding the individual digits as you go from left to right. When you reach the end of the number, if your final result is a multiple of 11, the original number will be divisible by 11.
  12. Another two-part test. Because 12 is a multiple of 3 and 4, if your number passes both of the divisibility tests for these two smaller numbers, then it will be evenly divisible by 12.

The divisibility rule for 7 is an example of the right-trim method, which may be applied in performing divisibility tests for larger values, provided you know what to multiply the right-most digit of your test number by in applying that process. In the case of 7, you can also apply it by multiplying the right-most digit of your number by 2 and subtracting that result from the remaining left-side of your number - there's more than one way you can make it work to find out if your test number is divisible by 7.

Going back to the number 1,512, hopefully, you found that it is divisible by 1, 2, 3, 4, 6, 7, 8, 9, and 12. If you're looking for a challenge, try 5,856,519,049,581,039. And if you are looking for more divisibility rules for larger numbers, keep reading....

Right Trim Multipliers

Now, what about divisibility rules for numbers bigger than 12? For most of these numbers, the right trim method provides a relatively simple and effective means to determine if these numbers can evenly divide into your number of interest. Provided you know what multiplier to apply in using it.

At the same time, you may also recognize from the rules we presented for divisors from 1 to 12 that you don't get a whole lot of extra mileage in having divisibility rules for conjugate numbers, or rather, those values that are already the products of two or more smaller factors. For example, you can simply perform the tests for 2 and 7 to determine if a number is wholly divisible by 14. Or you could perform the tests for 3 and 11 to determine if your number is divisible by 33.

From a practical perspective then, if you're trying to tell if a given value is divisible by a smaller number, you only need to apply the divisibility rules that apply for prime numbers, which you can combine together as you might need to conduct divisibility tests for non-prime number divisors.

That brings us to the following tool, which we've constructed to identify the multiplier that you would need to successfully apply the single digit right trim method with your number of interest. Just enter the prime number for which you want to identify a multiplier, and it will provide you the number you need.

Divisibility Rule Generator
Input Data Values
Prime Number

Divisibility Rule Multiplier
Calculated Results Values
'Single Digit' Right Trim Multiplier

Technically, the tool above will work to identify valid right trim method multipliers for any value you enter whose final digit is 1, 3, 7 or 9, which coincidentally happens to include all prime numbers greater than 5. Speaking of which, here are lists of the first fifty million prime numbers, which should keep you busy for a while for your divisibility tests!

Image credit: unsplash-logoGayatri Malhotra

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December 12, 2019

The Federal Reserve's Open Market Committee (FOMC) concluded its two-day December 2019 meeting on 11 December 2019, and tried very hard to give the impression that it was done adjusting short term interest rates in the U.S. for now, holding the Federal Funds Rate in its current target range of 1-1/2 to 1-3/4 percent after having reduced it several times earlier in the year.

Combined with a generally rising spread in the U.S. Treasury yield curve, the lowered Federal Funds Rate has reduced the probability that a new economic recession will someday be determined to have started in the United States during the twelve months from December 2019 to December 2019. Through 11 December 2019, those odds now stand at 1 in 18, which rounds down to roughly a 5% probability.

U.S. Recession Probability Track Starting 2 January 2014, Ending 11 December 2019

Those odds had previously peaked at one in nine back on 9 September 2019, where if the NBER eventually NBER ever does determine that the national U.S. economy entered into recession at some future time, they will most likely identify a month between September 2019 and September 2020 as its starting date.

These probabilities come from a recession forecasting method developed back in 2006 by Jonathan Wright, which uses the level of the effective Federal Funds Rate and the spread between the yields of the 10-Year and 3-Month Constant Maturity U.S. Treasuries to estimate the probability of recession based on historical data.

With the rolling one-quarter average of the spread between the 10-Year and 3-Month U.S. Treasuries having turned positive, we will be reducing the frequency at which we will provide updates to this series to coincide with the FOMC's meeting schedule, which are held at approximately six week intervals. Should events cause the Treasury yield curve to re-invert however, with the yield on the 10-Year constant maturity U.S. Treasury falling below the yield of the 3-Month constant maturity U.S. Treasury, we will resume more frequent updates.

But you don't have to wait for us to analyze the recession odds for yourself! Just take advantage of our recession odds reckoning tool, which is really easy to use. Plug in the most recent data available, or the data that would apply for a future scenario that you would like to consider, and compare the result you get in our tool with what we've shown in the most recent chart we've presented above to get a sense of how the recession odds are changing.

As we get set to close the books on 2019, there are several hanging risks that could prompt such a change, with the potential expansion of the Fed's current QE-like effort into a full-on quantitative easing program to tame a liquidity crisis that has developed in repurchase "repo" markets over the last several months leading the list.

Previously on Political Calculations

We've been tracking the ebb and flow of heightened recession odds since June 2017 - here are all the posts in our latest recession forecasting series!


December 11, 2019

Four weeks ago, we took a snapshot of an improving outlook for the dividends investors expect to be paid out in each quarter of 2020. One month later, the CME Group's crystal ball for quarterly S&P 500 dividend futures indicates that those expectations have continued to rise.

The following animated chart shows how the future has changed since 10 September 2019, when we took our first snapshot of the expected future for the S&P 500's dividends in 2020. If you're accessing this article on a site that republishes our RSS news feed and you don't see the chart change every 3-4 seconds, please click through to our site to see the animation below.

Animation: Past and Projected Quarterly Dividends Per Share Futures for S&P 500, 2018-Q1 Through 2020-Q4, Snapshots on 10 September 2019, 6 November 2019, and 10 December 2019

Most of the change has occurred over the last seven weeks, since 21 October 2019, with the bulk of the change taking place between 21 October 2019 and 6 November 2019. Since then, the S&P 500's quarterly dividend futures have continued to rise, though at a slower pace. Links to our previous analysis of the future for the S&P 500's dividends through all of 2020 are below....

Previously on Political Calculations

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December 10, 2019

In October 2018, the value of China's exports to the United States reached an all time record high of $52.2 billion. Although the tariff war between the two nations had begun in earnest six months earlier, Chinese exporters raced to beat even higher tariffs that would take effect in 2019, artificially inflating the country's exports figures in the final months of 2018.

One year later, measured against that unusually high mark, the value of China's exports to the United States has crashed thanks to those higher tariffs. The total value of China's exports to the U.S. was $40.1 billion in October 2019, a 23.1% year over year decline.

Value of U.S. Imports from China, January 1985 - October 2019

The large percentage decline in the value of China's exports to the U.S. is very evident in our chart tracking the exchange rate-adjusted, year-over-year growth rate of the two nations' exports to each other.

Year Over Year Growth Rate of Exchange Rate Adjusted U.S.-China Trade in Goods and Services, January 1986 - October 2019

Curiously, U.S. exports to China are only slightly lower in October 2019 than they were a year earlier, which is attributable to China's strategy of targeting U.S. soybeans early in the tariff war. China's tariffs on U.S. soybeans in 2018 led Chinese importers to effectively boycott the U.S. crop that year, where a year later, very little additional negative effect is being observed from what might be considered to be China's most effective retaliatory tariff to date.

That's despite China waiving its tariffs on both U.S.-produced soybeans and pork in its attempt to undo some of the self-inflicted damage related in part to some of the unintended consequences of its tariff war strategy. Unfortunately, the nature of that damage is such that China's short-term demand for soybeans has been greatly diminished, with at least 41% of its domestic hog population lost to African Swine Fever. China won't need to import the kind of quantities of soybeans it was prior to the tariff war for several years, where it may be able to get by with soybeans exported from the world's second and third-largest soybean producers, Brazil and Argentina.

Our final chart shows how large the combined loss of trade between the U.S. and China is when measured against the counterfactual for how great it could be, if not for the tariff war.

Combined Value of U.S. Exports to China and Imports from China, January 2008 - October 2019

Here's the takeaway comment from the chart: In October 2019, the gap between the pre-trade war trend and the trailing twelve month average of the value of goods exchanged between the U.S. and China expanded to $14.2 billion. The cumulative gap since March 2018 has grown to $101.2 billion. As you can see in the chart, the magnitude of actual trade losses from 2018 to 2019, as measured by the rolling 12-month average indicated by the heavy black line, exceeds the actual trade losses that were recorded from 2008 to 2009 during the Great Recession.


December 9, 2019

The S&P 500 (Index: SPX) continued its record-setting spree in the first week of December 2019, ending the week at a new all-time record closing value of 3,145.91.

Although some market observers are claiming the S&P 500's rising trend is attributable only to rising valuations, with the index' Price/Earnings (P/E) ratio reaching up to its highest levels since 2001 without a proportionate increase in earnings, that's missing the point that earnings aren't what drive stock prices. Expectations for future dividends are what do, and they've been rising throughout much of 2019-Q4, with stock prices coming along for the ride.

As of the close of trading on Friday, 6 December 2019, we find the level of the S&P 500 is right in the middle of the range of the redzone forecast provided by our dividend futures-based model would set the index, assuming that investors are focusing on the distant future quarter of 2020-Q3 as they go about setting current day stock prices.

Alternative Futures - S&P 500 - 2019Q4 - Standard Model with Redzone Forecast Focused-on-2020Q3-Between 26-Nov-2019 and 07-Jan-2020 - Snapshot on 6 Dec 2019

2019-Q3 has become a focal point for investors because that's when investors are anticipating the Federal Reserve's next action on interest rates, with the CME Group's FedWatch Tool currently projecting at least a quarter point rate cut taking place in September 2020:

CME Group FedWatch Tool Probabilities of Federal Funds Rate Changing at Future FOMC Meeting Dates, Snapshot on 06 December 2019

The FedWatch tool is also now projecting another quarter point rate cut taking place before the end of 2020, but it wouldn't take much to move the probabilities enough to generate the anticipation of a half point rate cut taking place during 2020-Q3.

That's not to say that investors will maintain their focus on that distant future quarter over all the time between now and its end. They could shift their attention toward another quarter in the future, such as 2020-Q2, which would address the market analysts concerns about high P/E ratios for the S&P 500, since that change in how far into the future investors are looking would be accompanied by a fall in stock prices.

Such a change would be driven by the onset of new information. Speaking of which, here are the major headlines we noted for their market moving potential during the week that was.

Monday, 2 December 2019
Tuesday, 3 December 2019
Wednesday, 4 December 2019
Thursday, 5 December 2019
Friday, 6 December 2019

Over at The Big Picture, Barry Ritholtz succinctly summarizes the positives and negatives he found in the week's economics and market-related news.

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