Multiples and Divisibility

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1 Multiples and Divisibility A multiple of a number is a product of that number and an integer. Divisibility: A number b is said to be divisible by another number a if b is a multiple of a. 45 is divisible by 5 because 45 is a multiple of 5 (45 = 9 5) 20 is divisible by 4 because 20 is a multiple of 4 (20 = 5 4) Saying that b is divisible by a means that b a results in a remainder of zero. When this happens, we sometimes say that a divides b evenly. Example: 1. Determine whether 138 is divisible by Since the remainder is not 0, we know that 138 is not divisible by 4. Divisibility Rules We will examine divisibility rules for 2, 3, 5, 6, 9 and 10. Divisibility by 2: A number is divisible by 2 (is even) if it has a ones digit of 0, 2, 4, 6, or 8 (that is, it has an even ones digit). Determine whether each of the following numbers is divisible by is NOT divisible by 2, because the last digit is odd is divisible by 2, because the last digit is even is divisible by 2, because the last digit is even.

2 Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. Determine whether each of the following numbers is divisible by is divisible by 3, because the sum of the digits is = 9 is divisible by 3. (9 3 = 3) is divisible by 3, because the sum of the digits is = 12 is divisible by 3. (12 3 = 3) is NOT divisible by 3, because the sum of the digits is = 14 is not divisible by 3. Divisibility by 5: A number is divisible by 5 if its ones digit is 0 or 5. Determine whether each of the following numbers is divisible by is divisible by 5, because the last digit is ,488 is NOT divisible by 5, because the last digit is not 0 or ,200 is divisible by 5, because the last digit is 0. Divisibility by 6: A number is divisible by 6 if its ones digit is 0, 2, 4, 6, or 8 (is even) and the sum of its digits is divisible by 3. Determine whether each of the following numbers is divisible by , , is NOT divisible by 6, because the last digit is odd. Thus, it is not divisible by ,488 is NOT divisible by 6, because the sum of the digits = 23 is not divisible by 3. Although, it is divisible by ,200 is divisible by 6, because the last digit is even and the sum of the digits = 6 is divisible by 3.

3 Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9. Determine whether each of the following numbers is divisible by is divisible by 9, because the sum of the digits = 18 is divisible by ,488 is NOT divisible by 9, because the sum of the digits = 23 is not divisible by ,200 is NOT divisible by 9, because the sum of the digits = 5 is not divisible by 9. Divisibility by 10: A number is divisible by 10 if its ones digit is 0. Determine whether each of the following numbers is divisible by is NOT divisible by 10, because the last digit is not ,488 is NOT divisible by 10, because the last digit is not ,200 is divisible by 10, because the last digit is 0.

4 Factorizations Factors and Factorizations: A number c is a factor of a if a is divisible by c. A factorization of a expresses a as a product of two or more numbers. 1. Find all factors of Find all factorizations of The factors of 24 are all of the divisors of 24. Factors of 24: 1, 2, 3, 4, 6, 8, 12, The factorizations of 24 are as follows: 1 24, 2 12, 3 8, and 4 6 Prime and Composite Numbers: A natural number that has exactly two different factors, itself and 1, is called a prime number. The number 1 is not a prime number. A natural number, other than 1, that is not prime is composite. Determine whether the numbers listed below are prime, composite, or neither is divisible by 2. Therefore, it is composite. 13 is divisible by only one and itself. Therefore, it is prime. 24 is divisible by 2. Therefore it is composite. 89 is divisible by only one and itself. Therefore, it is prime. List of Primes from 2 to 100 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

5 Prime Factorization: Each composite number has a unique prime factorization. The prime factorization of a number is the product of factors where each factor is a prime number. Find the prime factorization for each of the following numbers Solutions: is divisible by 2. So, we have The number 42 is also divisible by 2. Now, we have The number 21 is divisible by 3. This gives us Since the factors above are all prime numbers, we have found the prime factorization for = or is divisible by 3. So, we have The number 100 is divisible by 2. Now, we have The number 50 is also divisible by 2. This gives us The number 25 is divisible by 5. So, we have Since the factors are all prime numbers, we have found the prime factorization for = or

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