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After the most successful and very useful article on THE ULTIMATE GUIDE TO NUMBER CLASSIFICATION, LazyMaths.com is pleased to offer another path breaking article – THE ULTIMATE COLLECTION OF EVERY KNOWN TESTS OF DIVISIBILITY.
As a part of any speed math exam like GMAT, GRE or CAT, if you are trying to find out whether that number is evenly divisible without actually dividing it, then use these tests of divisibility. Learn these tests and save yourself precious time.
Divisor
Divisibility condition
Examples
1
Automatic
Every number is divisible by 1
2
The last digit is even (0,2,4,6 or 8 )
1294: 4 is even
3
The sum of digits is divisible by 3.
405: 4 + 0 + 5 = 9, which clearly is divisible by 3.
4
The number formed by the last two digits is divisible by 4.
40832: 32 is divisible by 4.
5
The last digit is 0 or 5.
490: the last digit is 0.
6
It is divisible by 2 and 3.
1,458: 1 + 4 + 5 + 8 = 18, 1 + 8 = 9, so it is divisible by 3 and the last digit is even, hence number is divisible 6.
7
If you double the last digit and subtract it from the rest of the number and the answer is divisible by 7.
483: 48 – (3 × 2) = 42 = 7 x 6.
8
The number formed by the last three digits is divisible by 8.
56: (5 × 2) + 6 = 16.
9
The sum of digits is divisible by 9.
2,880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9.
10
The number ends in 0.
130: the last digit is 0.
11
If you sum every second digit, and subtract all the other digits, the answer is divisible by 11.
918,082: 9 – 1 + 8 – 0 + 8 – 2 = 22.
12
The number is divisible by 3 and 4.
324: (32 × 2) − 4 = 60.
13
Add 4 times the last digit to the rest, the answer is divisible by 13.
637: 63 + (7 × 4) = 91, 9 + (1 × 4) = 13.
14
It is divisible by 2 and 7
224: it is divisible by 2 and by 7.
15
It is divisible by 3 and 5
390: it is divisible by 3 and by 5.
16
Sum the number with last two digits removed, times 4, plus the last two digits.
176: (1 × 4) + 76 = 80.
17
Subtract 5 times the last digit from the rest, the answer is divisible by 17.
221: 22 – (1 × 5) = 17.
18
It is divisible by 2 and 9.
342: it is divisible by 2 and by 9.
19
Add twice the last digit to the rest, the answer is divisible by 19.
437: 43 + (7 × 2) = 57.
20
The number formed by the last 2 digits is divisible by 20.
The number formed by the last two digits is divisible by 25.
134,250: 50 is divisible by 25.
27
Since 37×27=999; the multiplier is one, taking three digits at-a-time. Sum the digits in blocks of three from right to left.
2,644,272: 2 + 644 + 272 = 918.
Subtract 8 times the last digit from the rest.
621: 62 − (1×8) = 54.
29
Add three times the last digit to the rest.
261: 1×3=3; 3+26= 29
31
Subtract three times the last digit from the rest.
32
The number formed by the last five digits is divisible by 32, as follows:
If the ten thousands digit is even, examine the number formed by the last four digits.
41,312: 1312.
If the ten thousands digit is odd, examine the number formed by the last four digits plus 16.
254,176: 4176+16 = 4192.
Add the last two digits to 4 times the rest.
1,312: (13×4) + 12 = 64.
33
Add 10 times the last digit to the rest.
627: 62 + 7 x 10 = 132, 13 + 2 x 10 = 33.
37
Sum the digits in blocks of three from right to left. Since 37×27=999; round up to 1000; drop the three zeros; the multiplier is one, taking three digits at-a-time. Add these products, going from right to left. If the result is divisible by 37, then the number is divisible by 37.
The New York Times recently (Sep 16) published an interesting article – “Gut Instinct’s surprising role in Math”. The article shares an interesting view about human mind’s ability to instinctively solve math problems. It shows how the mind is wired to perform some seemingly complex math tasks with ease.
To quote it from the article “Whenever we choose a shorter grocery line over a longer one, or a bustling restaurant over an unpopular one, we rally our approximate number system, an ancient and intuitive sense that we are born with and that we share with many other animals. Rats, pigeons, monkeys, babies — all can tell more from fewer, abundant from stingy. An approximate number sense is essential to brute survival: how else can a bird find the best patch of berries, or two baboons know better than to pick a fight with a gang of six?” It’s interesting to know that not only the human brain works this way but even baboons’ and rodents’ work in the same manner.
In fact, research has shown that new born babies if found to be good at deciphering approximations with ease, will carry on the same trait when they grow up and have to deal with abstruse mathematical problems. NYT states: “One research team has found that how readily people rally their approximate number sense is linked over time to success in even the most advanced and abstruse mathematics courses. Other scientists have shown that preschool children are remarkably good at approximating the impact of adding to or subtracting from large groups of items but are poor at translating the approximate into the specific.” In fact, it goes further and suggests that Math teachers should emphasize on the power of approximation early on in the child’s education so he can hone this natural skills to an advantage. To quote “Taken together, the new research suggests that math teachers might do well to emphasize the power of the ballpark figure, to focus less on arithmetic precision and more on general reckoning.”
At LazyMaths.com, we precisely try to encourage our students to leverage these skills of approximation. We have found that with practice, the line between approximation and precision can be diluted if not fully erased. In the section of Don’t Solve, LazyMaths offers a large number of examples of a variety of Math problems attempted using the techniques of approximation.
A lot of mathematics exams nowadays, e.g. GMAT, SAT, etc, are either completely multiple choice or they have multiple-choice components. These exams required students to solve a stipulated large number of questions within a small amount of time. Students break a lot of sweat over solving these questions, by trying to solve them completely like other questions, and not utilizing the information that is in front of them.
One of the basic difference between multiple-choice questions and other questions is that for multiple-choice ones you have to just figure out the answer, and for achieving this objective, you need not necessarily solve the question completely, as the answer is right in front of you, among the 4-5 choices that have been provided.
Ball-park, or possible range strategy, is one such very effective method to quickly solve a multiple-choice question in arithmetic. In order to use it, you first make a quick guess of what the possible range of your answer will be. Once, you have made such a guess, you can easily eliminate all the other answers, which don’t fall in that range, and so can be ruled out as a choice for the possible answer. You are now left with just 1 or 2 choices, which you can just plug-in to confirm which one is the correct answer.
This strategy doesn’t work well if all the choices are in the possible range, and it is particularly useful when the answers are scattered over a large range, as in the following example:
If 0.303z = 2,727, then z =
a) 9,000
b) 900
c) 90
d) 9
e) 0.9
We can notice that the range of answers is too large, so we can use ball parking to solve this question. .303 is very close to 1/3, which means 1/3 of z = 2,727, then what answer could be possibly correct. You don’t even have to do the math. 2,727 is about 1/3 of 9,000; therefore, the answer must be 9,000, according to the Ballpark Strategy (note that there are no other answers even in the 9,000 range.
If you don’t use the ballpark strategy, you could multiply both sides by 1000 to eliminate the decimal points, and then divide 2,727,000 by 303 and get the same answer, although after spending much more time.
Another important use of this strategy is to double-check your answer. Once you have solved a multiple-choice arithmetic question, you should confirm whether the answer lies in the ballpark of what the answer could be.
So, invest sometime in learning and practicing to use ball-parking. This will allow you to solve the arithmetic questions much faster and will save you precious time, enabling you to score much higher.
LazyMaths.com focuses on this strategy and has hundreds of examples to show how and where one can use ballparking. Visit the Don’t Solve section of the site to know more about it and check out some free examples.
This post is part of LazyMaths’ Math website review series. Dedicated purely to help math students, the review highlights tangible learning that the website offers to its visitors and users. If you would like to have your website considered for review, drop us an email – contact(at)lazymaths(dot)com.
Review in a Tweet
Comprehensive set of interactive math lessons, with unlimited practice
About the website AAA math features a comprehensive set of interactive math lessons, categorized well in terms of levels (Kindergarten through Eighth) or according to topics, like Addition, Algebra, Counting, Decimals, etc. This allows the user to use the site for class related work, as well as reviewing or learning any particular topic that he or she needs.
The most striking part of the website is the focus on interactivity. Fully interactive lessons enable illustration of key concepts through visual animation, reinforcement of important points through challenging fun math games, and prevention of learning of wrong ways of solving through immediate feedback. Also, interactivity engages the students, and makes learning math a fun activity.
Moreover, availability of unlimited practice questions allows the students to try out enough questions till they feel confident in a non-threatening environment. This has a huge impact on their self-esteem and confidence.
The content is available in two languages “English” as well as “Spanish”, allowing for a wider reach. Also, all the website content along with some additional stuff is available on a CD version. The CD version is completely advertisement free. It helps the students in avoiding distractions, and focus completely on learning.
What’s on the website?
AAAmath offers the following:
• A wide range (Kindergarten through Eighth) of lessons
• Unlimited practice questions on each topic
• Math Games
• Immediate feedback to the students
• Links to other mathematics resources
What’s the best part about this website?
No registration is required, and the site is very well structured, making navigation intuitive.
Any $$$?
Only if you buy the CD version – ranges from $24.95 to $240.95 depending on Home, Classroom or School editions
Review Disclaimer: This is an independent review of the given site. It only reflects the opinion of our site reviewer(s). Views and opinions expressed may not be representative of the site or its owners or its users and visitors. LazyMaths.com and its reviewer(s) cannot be held responsible for any damage caused to hardware or any other problem caused by visiting this site or misuse of this information. Any errors or omissions should be brought to our attention by contacting the site administrator.
This post is part of LazyMaths’ Math website review series. Dedicated purely to help math students, the review highlights tangible learning that the website offers to its visitors and users. If you would like to have your website considered for review, drop us an email - contact(at)lazymaths(dot)com.
Mathkinz.com – Rule the World (of Math)
Review in a Tweet –
By students for students – Download Math practice sheets for free from Mathkinz.com
About the website – Mathkinz is a unique site that’s entirely run and managed by kids. Yup, school kids of grades 4 through 8. They develop the content, in this case worksheets, test material, etc. and host it too!
The website was started by Nirlesh Jain, a father that takes keen interest in his kids’ education. While teaching his kids, he came up with the idea. After speaking to his friends and family, he gathered all the neighboring kids and encouraged them to start developing math content. And in just a few months, Mathkinz.com was born. To keep the interests of kids alive, he offers Mathkinz points for the content they develop. This encourages them to keep developing more content and also build up a competitive spirit amongst them.
Mathkinz also conducts Math competitions in the central New Jersey area to encourage local students to develop better math skills.
What’s on the website?
Mathkinz offers the following:
• An online Math Quiz (By Grades – 1 through 8 – US Standards)
• Math Practice sheets (By Grades – 1 through 8 & By Subjects )
• Math Jokes
• Math Games
• Math Resources (Other math websites links)
• Math News
• Math Forum (for parents to network, ask questions, provide suggestions, etc.)
What’s the best part about this website?
You don’t have to register or sign up. Just visit the site and start printing your Math worksheets at home for your kids!
Any $$$?
Mathkinz is free, yes really!!
Review Disclaimer: This is an independent review of the given site. It only reflects the opinion of our site reviewer(s). Views and opinions expressed may not be representative of the site or its owners or its users and visitors. LazyMaths.com and its reviewer(s) cannot be held responsible for any damage caused to hardware or any other problem caused by visiting this site or misuse of this information. Any errors or omissions should be brought to our attention by contacting the site administrator.
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We have been using numbers in everyday life. Everything from 0 to 22/7 might sound the same to most, but numbers differ from one another. Based on their characteristics, they are classified in groups. Based on all the different types of numbers Mathematician’s have named, we have built the ultimate guide to Number classification.
The associated chart (at the bottom of the article) shows how the number groups are related to each other. Read below for more details on each number group.
Real Numbers – All kinds of numbers that you usually think of – from bus route numbers, to your weight, to pi and even the square root of pi! In short everything!! Everything? Really? Well…almost
Imaginary numbers – Have you ever tried finding the square root of -1? If you haven’t, try it on your calculator. It might show an error (if it is a dumb calc) or it might show an ‘i’. That little ‘i’ is called an imaginary number. In short square roots of negative numbers make imaginary numbers.
Complex Numbers – It’s rather simple! Make a combination of Real and Imaginary numbers and voila! You get a Complex number. Stuff like 3+2i or 3/4i make up complex numbers. Just think of it when you mix a real number with an imaginary one, things do get a bit complex!
Rational Numbers – Any number that can be written as a fraction is a rational number. So numbers like ½, ¾, even 22/7 and all integers are also rational numbers.
Irrational Numbers – Simply the opposite of rational numbers i.e. numbers that cannot be written as fraction, like square roots of prime numbers, the golden ratio, the real value of pi (22/7 is a mere approximation not the real value of pi) are irrational numbers.
Integers – Any number that is not a fraction and does not have a tail after the decimal point is an integer. This includes both negative as well as positive numbers as well as zero.
Fractions – Numbers that are expressed in a ratio are called fractions. This classification is based on the number arrangement and not the number value. Remember that even integers can be expressed as fractions – 3 = 6/2 so 6/2 is a fraction but 3 is not.
Proper Fractions – Whenever the value of the numerator in a fraction is less than the value of the denominator, it is called a proper fraction. i.e. it’s bottom heavy.
Improper Fractions – Whenever the value of the denominator in a fraction is less than the value of the numerator, it is called a proper fraction. i.e. it’s top heavy.
Mixed Fractions – All improper fractions can be converted into an integer with a proper fraction. This combination of an integer with a proper fraction is called a mixed fraction.
Natural Numbers – All positive integers (not including the zero) are Natural numbers. Simply put, whatever you can count in Nature uses a natural number.
Whole Numbers – All positive integers inclusive of the zero are Whole numbers. Not a big deal different from Natural numbers.
Even Numbers – All integers that end with a 0, 2, 4, 6, or 8 (including the numbers 0, 2, 4, 6 & 8 themselves) are even numbers. Note that ‘0’ itself is an even number. Also note that negative numbers can also be even so long as they can be integrally divided by 2.
Odd Numbers – All integers that are not even numbers are odd number
Prime Numbers – A natural number, more than one, which has exactly two distinct natural number divisors: 1 and itself – is called a Prime number. There can be infinite prime numbers.
Composite Numbers – A positive integer which has a positive divisor other than one or itself is a composite number. In other words, all numbers that are not prime are composite
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