Grade 6 Math Circles March 1-2, Introduction to Number Theory
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1 Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles March 1-2, 2016 Introduction to Number Theory Being able to do mental math quickly is such an important skill in life, so even when you don t have a calculator on you, you are still able to calculate whatever you need, such as if you re at the mall or out for lunch. Today we re gonna go over some tricks for quick multiplication and checking for divisibility. Multiplication Tricks Number Trick Example 2 Add the number to itself [2 n = n + n] 2 34 = = The product will either end in 0 or 5. Also equivalent to dividing it by 2 and multiplying by 10. If the number is even, the product will end with the same digit. For n 9, on your ten fingers, put down your n th finger. The number of fingers before the n th is the first digit of the product, and the number of those after it is the second digit. Also, the sum of the digits will be = = 10 4 = = 12, 6 12 = Just place a zero at the end of the number = For n < 10, just repeat n twice. For 10 n < 100, place the sum of the digits in between the 2 digits. If the sum is greater than 9, carry over the 1. Multiply it by 10, then add 2 n (ie. 12 n = (10 n) + (2 n)) = 1(1 + 5)5 = 165 and = 3(3+8)8 = 3(11)8 = (3 + 1)18 = = (10 5) + (2 5) = = 60 1
2 Divisibility Tricks Definition: An integer x is divisible by an integer n if x n is an integer (ie. there is no remainder). Note: Saying x is divisible by n is the same as saying Num Trick Example n is divisible by 2 if the last digit of n is even (ie. 0, 2, 4, 6 or 8). n is divisible by 3 if the sum of its digits is divisible by 3. n is divisible by 4 if the number formed by the last 2 digits is divisible by is divisible by 2, but isn t is divisible by 3 since = 24 and 24 is divisible by is divisible by 4 since 32 is divisible by 4. 5 n is divisible by 5 if the last digit is 0 or is divisible by 5 since it ends in a n is divisible by 6 if it is divisible by 2 and 3 (so check both tests). n is divisible by 7 if when you subtract 2 times the units digits from the rest, repeatedly, the outcome is divisible by 7. n is divisible by 8 if the number formed by the last 3 digits is divisible by 8. n is divisible by 9 if the sum of its digits is divisible by is divisible by 6 since the last digit is even and = 33 which is divisible by (7) = (2) = (2) = 14 since 14 is divisible by 7, then so is is divisible by 8 since 880 is divisible by is divisible by 9 since = 36 is divisible by n is divisible by 10 if the last digit is is divisible by 10 since the last digit is n is divisible by 11 if the alternating sum of its digits is divisible by 11. (Alternating means you add one and subtract the next and so on.) is divisible by 11 since = 11 which is clearly divisible by n is divisible by 12 if it is divisible by 3 and is divisible by 12 since 56 is divisible by 4 and = 18 is divisible by 3. 2
3 Number Theory You may be thinking to yourself, like so what? How is knowing whether a number is divisble by another number ever going to help me? There are actually many applications to this. So examples include: 1. Splitting a deck of cards among a specific number of people, it would be helpful to know what numbers divide 52 and which numbers do not. 2. Splitting a number of pizzas or pies among a number of people, it would be helpful to know how many slices you should slice up each one into. 3. Finding the greatest common divisors of 2 or more numbers. Greatest Common Divisors Today, we re going to do a few examples where we find the GCD of 2 numbers using prime factorization. Do you remember what a prime number is? It is basically any number whose divisors are only 1 and itself, or in other words it is not a multiple of any other number. So in this section all we will do is write really big numbers in terms of a product of prime numbers. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,... Example (a) Find the GCD of 231 and 660 (c) Find the GCD of 442 and 289 So, gcd(231, 660) = 3 11 = 33 (b) Find the GCD of 1386 and 322 (d) Find the GCD of 480 and
4 Coprimes Remember that primes are number whose only factors are 1 and itself. Similarily, 2 integers are coprimes if the only integer which evenly divides both of them is 1 Think of it as common factors. The only common factor is 1. For example, 4 and 9 are coprimes since the prime factorization of 4 and 9 are 4 = 2 2 and 9 = 3 3 respectively, so they have no common factors. Connection: What should the GCD come out to be if they are coprime? Example 5 Determine which of the following pairs of numbers are coprimes. (a) 15 and 18 (b) 19 and 21 (c) 16 and 25 (d) 36 and 45 (e) 46 and 55 (f) 6 and 25 (g) 12 and 21 (h) 16 and 27 (i) 15 and 28 (j) 30 and 42 (k) 105 and 42 4
5 Problem Set 1. Calculate each of the following products (without a calculator): (a) 9 8 = (b) 6 8 = (c) 9 5 = (d) 6 11 = (e) = (f) = (g) = (h) 36 9 = (i) 36 4 = (j) = (k) 16 7 = (l) 17 5 = (m) 6 8 = (n) = (o) 8 11 = (p) = (q) = (r) = (s) 17 4 = (t) = 2. Determine the GCD of the following numbers. (a) gcd (435, 377) (b) gcd (273, 595) (c) gcd (9081, 3270) (d) gcd (36, 150) (e) gcd (442, 289) (f) gcd (144, 720) (g) gcd (60, 1764) (h) gcd (204, 22050) (i) gcd (11550, 36) 5
6 3. Use the divisiblity rules to circle all the correct answers. Number Divisible by: Example:
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