5S Online - Did You Solve It

Earlier today I posted the following video, in which I asked Google Assistant to calculate the factorial of 100.

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The factorial of 100 is the multiplication 100 x 99 x 98 x … x 3 x 2 x 1 in which 100 is multiplied by every whole number below it.

The answer is 158-digits long. Google Assistant’s valiant effort, however, does not get every digit correct.

Today’s puzzle was:

How many zeros does the factorial of 100 really have at the end of it?


I mentioned in the original post that if a number has a zero at the end of it, it is divisible by 10. What we need to do here is work out how many times 10 divides into the number 100 x 99 x 98 x … x 3 x 2 x 1.

Let’s start: 10 divides once each into 10, 20, 30, 40, 50, 60, 70, 80, 90 and twice into 100, which means there must be at least 11 zeros at the end of 100!.

Yet it is possible to multiply two numbers that don’t end in 0 to create one that does. For example, 8 x 5 = 40. How do we account for all the times that numbers in the breakdown of 100! multiply together to make a number divisible by 10?

The clue is to realise that 10 = 2 x 5. And that every time two numbers multiply together to create a number that is divisible by 10, there must be a 2 and a 5 involved.

E.g. 8 x 5 = 2 x 2 x 2 x 5 = (2 x 2) x (2 x 5) = 4 x 10 = 40.

So, we can rephrase our task as having to look for all the instances of (2 x 5) in 100!. In other words, we need to break down every number from 1 to 100 into its factors and see how many times 5 and 2 appear.

How many times does 5 appear as a factor in the numbers from 1 to 100? Well, counting upwards in 5s, we get 5, 10, 15, 20…90, 95, 100. These 20 numbers have five as a factor. In fact 25, 50, 75 and 100 have 5 as a factor twice. So the number of times that 5 appears as a factor is 24.

We can quickly see that 2 appears as a factor at least 24 times (just count the even numbers), so the total number of times (2 x 5) appears in 100! must be 24. The number of zeros at the end of 100! is 24.

For those who are interested, 100! in all its glory is:


I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

I’m the author of several books of puzzles, most recently the Language Lover’s Puzzle Book. I also give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.

Smoke from wildfires in Canada is moving hundreds of miles into the U.S., affecting air quality in many areas across the Lower 48. Explore the Fire & Smoke Map for more details.






What should I do?

Pollution is hazardous at these levels. Everyone should take steps to reduce their exposure when particle pollution levels are in this range.Staying indoors – in a room or building with filtered air – and reducing your activity levels are the best ways to reduce the amount of particle pollution you breathe into your lungs. Read on for more information on steps to help reduce your exposure to short episodes of high levels of PM2.5.

Who needs to take steps to reduce exposure when PM2.5 levels are “hazardous” on the AQI?

Everyone needs to take steps to protect themselves when pollution levels are “hazardous” and above. Some people are at higher risk from PM2.5 exposure. People most at risk from particle pollution exposure include those with heart or lung disease (including asthma and chronic obstructive pulmonary disease-COPD), older adults, and children. Research indicates that pregnant women, newborns, and people with certain health conditions, such as obesity or diabetes, also may be more susceptible to PM-related effects.Not sure if the heart disease category applies to you? People with heart disease includes all people with known coronary artery disease, ischemic heart disease, history of angina and/or heart attack, stent placement, by-pass operation, heart failure, ventricular arrhythmia, peripheral vascular disease, history of stroke, transient ischemic attack (TIA), or cerebrovascular disease. This group also includes older adults, because they are more likely to have undiagnosed cardiovascular disease, along with people with multiple risk factors for cardiovascular disease, such as high blood pressure, elevated cholesterol, smoking, and diabetes.Why are children more at risk? Children are more likely to be exposed to air pollution, because they often spend more time outdoors engaged in activity and play, and they breathe more air per pound of body weight than adults. They are more susceptible to the effects of air pollution, because their airways are still developing. In addition, children are more likely than adults to have asthma, which increases their risk.If you are in an at-risk group, if you have heart or lung disease, if you are an older adult, or if you have children, talk with your doctor in advance about when and whether you should leave the area or move to a location with better indoor air quality. When PM2.5 concentrations are high for a prolonged period of time, fine particles can build up indoors even though you may not be able to see them.

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If you are in an at-risk group, don’t wait until pollution reaches the “hazardous” category to take action to reduce your exposure. Air quality is unhealthy for you when particle pollution levels reach the “unhealthy for sensitive groups” range, so you will need to take steps to reduce your exposure earlier and more often. If you are healthy, begin taking steps when air pollution reaches the “unhealthy” category.

How can I tell if particle pollution is affecting me?

Even if you are healthy, you may experience temporary symptoms such as irritation of the eyes, nose, and throat; coughing; phlegm; chest tightness; and shortness of breath. These symptoms should go away when air quality improves.If you have lung disease – including asthma and COPD – you may not be able to breathe as deeply or as vigorously as normal, and you may experience coughing, chest discomfort, wheezing, shortness of breath, and unusual fatigue. Make sure you follow your doctor"s directions about taking your medicines and following your asthma management plan. If you have any of these symptoms, reduce your exposure to particles and follow your doctor"s advice. Contact your doctor if symptoms persist or worsen. In the event of an emergency, anywhere in the U.S., dial 911.If you have heart or vascular disease, particle exposure can cause serious problems – including worsening of your disease – in a short period of time. Do not assume that you are safe just because you do not have symptoms.Symptoms that may indicate a serious heart problem include: Chest discomfort (uncomfortable pressure, fullness, squeezing, or pain in the center of the chest that lasts more than a few minutes or goes away and comes back), discomfort in other areas of the upper body (pain or discomfort in one or both arms, the back, neck, jaw or stomach), shortness of breath, or other signs may include breaking out in a cold sweat, nausea or light-headedness. Seek emergency medical treatment if you experience these symptoms. In the event of an emergency anywhere in the U.S., dial 911.Symptoms of a stroke include: Sudden numbness or weakness (in the face, arm or legs especially on one side of the body), confusion, trouble speaking or understanding, problems seeing in one or both eyes, dizziness, loss of balance or coordination or trouble walking, or severe headache with no known cause may indicate symptoms of a stroke. Seek emergency medical treatment if you experience these symptoms. In the event of an emergency anywhere in the U.S., dial 911.

What can I do to reduce my exposure to fine particle pollution when levels are extremely high?

Stay indoors in an area with filtered air. Particle pollution can get indoors, so consider purchasing an air cleaner if you live in an area with high levels of particle pollution. (See information on selecting an air cleaner below.)Air cleaners that remove particles include high-efficiency mechanical filters and electronic air cleaners, such as electrostatic precipitators. Avoid using an air cleaner that works by generating ozone, which will increase the pollution in your home.If you do not have air cleaners in your home, try to go somewhere that does have air filtration. This could, for example, be a friend’s home, if it has air filtration.Keep your activity levels low.Avoid activities that make you breathe faster or more deeply. This is a good day for indoor activities, such as reading or watching TV.If you cannot buy filters for your entire home, create a clean room for sleeping.A good choice is a room with as few windows and doors as possible, such as a bedroom.If the room has windows, keep them closed.Run an air conditioner or central air conditioning system if you are certain your air conditioner does not draw air from outdoors and has a filter. If the air conditioner provides a fresh air option, keep the fresh-air intake closed. Make sure that the filter is clean enough to allow good air flow indoors.Use an air filter in that room. Avoid using an air cleaner that works by generating ozone. Those types of cleaners will increase the pollution in your home.Follow steps for keeping pollution in your home low (see next section).Take additional steps to keep pollution in your home low. Air cleaners alone may not be enough. Because particle pollution from the outdoor air can easily get inside, take steps to avoid adding even more pollution indoors when outdoor PM2.5 levels are high:Avoid using anything that burns, such as wood fireplaces, gas logs and even candles or incense.Keep the room clean – but don’t vacuum unless your vacuum has a HEPA filter. That stirs up particles already inside your home. Wet mopping can help reduce dust.Don’t smoke.Be cautious when the weather is hot. If it is too hot to stay inside with the windows closed, or if you are in an at-risk group, go somewhere else with filtered air.When air quality improves, open the windows and air out your home or office.Selecting an air cleaner:Air cleaners that remove particles include high-efficiency mechanical filters and electronic air cleaners, such as electrostatic precipitators. Avoid using an air cleaner that works by generating ozone, which will increase the pollution in your home.Should I wear a dust mask if I have to go outside?Do not rely on dust masks for protection. Paper "comfort" or "dust" masks are designed to trap large particles, such as sawdust. These masks will not protect your lungs from small particles such as PM2.5. Scarves or bandanas won’t help either.Disposable respirators known as N-95 or P-100 respirators will help if you have to be outdoors for a period of time. It’s important that you wear the respirator correctly, however. For information on how to use one, see:

How will I know when conditions are better?

Also pay attention to weather forecasts; these can help you plan your activities for times when air quality improves, such as when winds are forecast that clear the air. When the air clears, and AQI readings are low, take advantage of these times to get outdoors.

Information sources:

Wildfire Smoke, A Guide for Public Health Officials: This document, originally developed by the California Air Resources Board (CARB) and the California Department of Public Health (CDPH), is designed to help local public health officials prepare for smoke events, to take measures to protect the public when smoke is present, and communicate with the public about wildfire smoke and health. 

Now website information on wildfire smoke

CDC information: Protect yourself from wildfire smoke: https://www.cdc.gov/disasters/wildfires/smoke.html

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