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We’ll start with the easy stuff. Multiplying by 4: (1) Double the multiplicand you want to multiply 4 by (2) Double it one more time e.g.     8 * 4 = 32 8 * 2 = 16 16 * 2 = 32 Why does this work? 4 can be broken up into 2 * 2 8 * 4 = 32 8 * (2 * 2) = 32 Thanks to the associative property of multiplication, we can multiply factors in whatever grouping or order we chose and still get the same answer. We start by multiplying multiplicand we want to multiply 4 by 2 because this computation is easy for most people to do in their heads. (8 * 2) * 2 (16) * 2 We then multiply our product by the remaining multiplicand, which is 2. 16 * 2 = 32 Multiplying by 10: Stick a zero behind whatever number you wish to multiply 10 by 988 * 10 = 9880 Why does this work? Consider what we’re doing in terms of place value. When... read more

We’ll start with the easy stuff. Multiplying by 4: (1) Double the multiplicand you want to multiply 4 by (2) Double it one more time e.g.     8 * 4 = 32 8 * 2 = 16 16 * 2 = 32 Why does this work? 4 can be broken up into 2 * 2 8 * 4 = 32 8 * (2 * 2) = 32 Thanks to the associative property of multiplication, we can multiply factors in whatever grouping or order we chose and still get the same answer. We start by multiplying multiplicand we want to multiply 4 by 2 because this computation is easy for most people to do in their heads. (8 * 2) * 2 (16) * 2 We then multiply our product by the remaining multiplicand, which is 2. 16 * 2 = 32 Multiplying by 10: Stick a zero behind whatever number you wish to multiply 10 by 988 * 10 = 9880 Why does this work? Consider what we’re doing in terms of place value. When... read more

Welcome back to the school everyone! I hope you all had a great summer. For all those whose summer was maybe a little too great, maybe those who’ve forgotten even the basics, we’re going to take it all the way back to arithmetic a.k.a “number theory”. A review of number theory is a perfect place to start for many levels. Calculus and a lot of what you learn in pre-calculus is based on the real number system. When we use the word “number” we are typically referring to all real number. But how can numbers be “real”? You can’t touch the number 6 or smell 1,063. You can’t boil 1/2 or stick it in a stew. So what’s so real about real numbers? The simple answer is this: a real number is a point on a number line (1).                               -2.5        -1      0 ... read more

I am taking from The Official Hunter College High School Test: problem 76 on page 20. We read the following. In the expression below, each letter represents a one digit number. Where the same letter appears, it represents the same number in each case. Each distinct letter represents a different number. In order to make the equation true, what number must replace C?    AAA    AAB + ABC   2012 A great start is to decode each AAA, AAB, and ABC. It helps to look at this problem wholly; particularly we look at the leading sum on the left wall (of the same types). We glean that either: (1) A + A + A = 20, (2) A + A + A + 1 = 20 or (3) A + A + A + 2 = 20: its very important to remember that given three numbers each less than ten, the sum of them which is great, is at most 2 in the tens place. This means that each row can only donate a 1 or 2 to the next. We can conclude that our line is... read more

Let's start off by defining some rules.   1) please no one post answers directly on this post, rather send all answers to me via a message.  Comments will be deleted and the person will be disqualified from all future contests.   2) The first 5 people to respond correctly to this post will receive a free 1 hour tutoring session via the online platform in any subject that I am approved in.  I will respond back to your message explaining the correct answer, how to get that answer, whether you were correct, and, of course, the details in setting up your free session with me   3) If you are trying, but stumped...message me for a hint   4) Have fun!!   Now for the brainteaser!   Chicken McNuggets can be purchased in quantities of 6, 9, and 20 pieces.  What is the largest amount of McNuggets that can NOT be purchased, using these quantites?   Happy Holidays everybody!  I look forward...

In mathematics, we start with the natural numbers (or more simply the 'counting' numbers) and learn how to count, starting with 1 and moving up the positive number line. But something special about counting numbers is usually overlooked- primes.    Looking at the naturals, we have {1, 2, 3, 4, 5, 6, 7, ... }  all the way to infinity.    Now if you pick a number, any number, and analyze it, you can see its basic properties such as its factors or its multiples. Let's take the number 4 for example.    4 is a multiple of 2, which means it can be divided by a number other than itself and 1. We write "2|4" meaning "two divides four."    Obviously, most other numbers have these factors and are built on them. But there is a type of number that you may be familiar with but not realize its significance in mathematics. One of the most interesting things in mathematics, though a basic concept,... read more

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