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....

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.

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: Gayatri Malhotra

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.

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!

• The Return of the Recession Probability Track
• U.S. Recession Probability Low After Fed's July 2017 Meeting
• U.S. Recession Probability Ticks Slightly Up After Fed Does Nothing
• Déjà Vu All Over Again for U.S. Recession Probability
• Recession Probability Ticks Slightly Up as Fed Hikes
• U.S. Recession Risk Minimal (January 2018)
• U.S. Recession Probability Risk Still Minimal
• U.S. Recession Odds Tick Slightly Upward, Remain Very Low
• The Fed Meets, Nothing Happens, Recession Risk Stays Minimal
• Fed Raises Rates, Recession Risk to Rise in Response
• 1 in 91 Chance of U.S. Recession Starting Before August 2019
• 1 in 63 Chance of U.S. Recession Starting Before September 2019
• 1 in 54 Chance of U.S. Recession Starting Before November 2019
• 1 in 42 Chance of U.S. Recession Starting Before December 2019
• 1 in 26 Chance of U.S. Recession Starting Before February 2020
• 1 in 16 Chance of U.S. Recession Starting Before April 2020
• 1 in 14 Chance of U.S. Recession Starting Before April 2020
• 1 in 13 Chance of U.S. Recession Starting Before May 2020
• 1 in 12 Chance of U.S. Recession Starting Before June 2020
• 1 in 11 Chance of U.S. Recession Starting Before July 2020
• Odds of U.S. Recession Before August 2020 Rise to 1 in 10
• 1 in 10 Chance of U.S. Recession Starting Before August 2020
• What The Dickens Is Going On With Recession Indicators?
• 1 in 9 Chance of U.S. Recession Starting Before October 2020
• U.S. Recession Odds Peak, Begin to Recede
• After Peaking at 1-in-9 in September 2019, Odds of U.S. Recession Falls to 1-in-11
• Post Peak, Recession Odds Continue to Recede
• 1 in 18 Chance of U.S. Recession Starting Between Dec-2019 and Dec-2020

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.

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

• The Futures for S&P 500 Quarterly Dividends Per Share Through 2020
• Improved Outlook for Dividends in 2020 Powers S&P 500 Upward

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.

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.

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.

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.

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.

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:

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

• Warning signs for U.S. economy:
• U.S. oil output growth slows: just how much is anyone's guess
• U.S. factory sector contracts again in November: ISM
• New tariffs, old tariffs:
• Trump, citing U.S. farmers, slaps metal tariffs on Brazil, Argentina
• U.S. may increase tariffs after WTO rejects EU claims over Airbus
• China wants rollback of tariffs in phase one trade deal with U.S.-Global Times
• Bigger trouble developing in the Eurozone:
Euro zone factory activity shrank in November but worst may be over: PMI
• Bigger trouble developing in South America:
• Chile's economy posts biggest drop in decade as protests bite
• Argentine steel industry chamber says U.S. tariffs would hit production, employment
• Chinese stimulus gaining traction?:
• China's factory activity surprises with fastest expansion in three years: Caixin PMI
• U.S. stocks retreat on economy and trade jitters

Tuesday, 3 December 2019

• Oil steadies after slide on Trump's U.S-China trade comments
• Multi-front tariff wars and trade negotiations:
• Trump signals trade deal with China could be delayed until after 2020 election
• U.S. vows 100% tariffs on French Champagne, cheese, handbags over digital tax
• EU ready to respond to U.S. tariff threat on France: Le Maire
• Bigger stimulus developing in Japan:
• Japan preparing $120 billion stimulus package to bolster fragile economy
• Wall Street falls as trade hopes wane

Wednesday, 4 December 2019

• Oil jumps 4% on U.S. stockpiles drop; further OPEC output cuts seen
• U.S. service sector slows more than expected in November
• Trump to sign U.S.-Japan trade deal proclamation next week: trade representative
• Bigger trouble developing in Germany and China:
• German car industry expects more job cuts in 2020
• German services sector growth stays anemic: PMI
• China capex growth hits three-year low as weak economy, trade war drag
• Bigger stimulus developing in China:
• China to implement more policies to stabilize jobs amid growing pressure
• Wall Street bounces back on renewed trade optimism

Thursday, 5 December 2019

• Trade data and negotiations:
• U.S. trade deficit shrinks sharply; labor market tight
• U.S.-China trade talks are 'on track': Mnuchin
• Trump threatens trade action to spur NATO contributions
• Bigger trouble developing in the Eurozone:
• Euro zone growth curbed by trade, retail sales sharply down
• Bigger stimulus developing in China:
• World Bank adopts $1 billion-plus annual China lending plan over U.S. objections
• Wall St. barely gains as investors wait for trade progress

Friday, 6 December 2019

• Oil rises sharply this week as OPEC+ agrees on deeper output cuts
• Strong U.S. job growth showcases economy's resilience
• Returning General Motors workers seen boosting U.S. November payrolls
• Don't touch that dial! Traders see Fed on hold until after 2020 election
• Fed voters a less-hawkish bunch in 2020, lowering bar for U.S. rate cut
• Reading the tea leaves in U.S.-China trade negotiations:
• Trump will make final call on China tariffs, likes direction of talks: Kudlow
• China to waive tariffs on some U.S. soybeans, pork in goodwill gesture
• Bigger trouble developing in the Eurozone:
• German industry slump sparks renewed economic growth fears
• Wall St ends higher after jobs report, trade optimism

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