How many positive integers less than or equal to 2022 have an odd number of factors of 2
Factors of a number are defined as numbers that divide the original number evenly or exactly. The meaning of a factor is a whole number that can divide a greater number evenly. A factor cannot be a fraction. Each prime number will have only two factors, i.e. 1 and the number itself, whereas all composite numbers will have more than two factors, that include prime factors also. Examples are: Show
Factors of an algebraic expression are the values that can divide the given expression. For example, factors of 3xyz are 3, x, y, z, 3x, 3y, 3z, xyz and 3xyz. Learn factorisation of algebraic expressions in detail here. What are Factors of a Number?By definition of factors, we know, they are the values that divide the original number into equal parts or numbers. For example, if 9 is the factor of 81, then if we divide 81 by 9, we get: 81 ÷ 9 = 9 Hence, 9 divides 81 into 9 equal parts. Or 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 = 81 Point to remember: Fractions cannot be considered as factors for any number. Factors of Prime NumbersThe factors of a prime number are only two, 1 and the number itself. We cannot divide prime number by any other number, that can give a whole number. 1 is not a prime number, since it has only one factor. These prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc. Factors of Composite NumbersThe composite numbers are those numbers that have factors more than 2. The prime factors of composite numbers are those factors that are prime numbers such as 2, 3, 5, 7, 11, etc.
Factors of Square NumbersSquare numbers are those that produced when a number is multiplied by itself. It is represented as n x n = n2, where n is any integer. 2 x 2 = 22 = 4 3 x 3 = 32 = 9 5 x 5 = 52 = 25 10 x 10 = 102 = 100 The above examples prove that one of the factors of a square number is the value, that is square to produce the original number. Factors FormulasThere are basically three types of formulas considered for factors. They are:
Let us assume N is a natural number, for which we need to find the factors. If we convert N into the product of prime numbers by prime factorization method, we can represent it as; N = Xa × Yb × Zc where X, Y and Z are the prime numbers and a, b and c are their respective powers. Number of FactorsThe formula for the total number of factors for a given number is given by;
Sum of FactorsThe formula for the sum of all factors is given by;
Product of FactorsThe formula for the product of all factors is given by;
Example: Find the total number of factors of 90 along with sum and product of all factors. Solution: Write the prime factorization of 90 first. 90 = 2 × 45 = 2 × 3 × 15 = 2 × 3 × 3 × 5 90 = 21 × 32 × 51 Here, X = 2, Y = 3, Z =5 and a = 1, b = 2, c = 1 Therefore, total number of factors of 90 = (a +1)(b+1)(c+1) = (1+1)(2+1)(1+1) = 2 × 3 × 2 = 12 Sum of factors of 90 = [(21+1-1)/2-1] × [(32+1-1)/3-1] × [(51+1-1)/5-1] = (3/1) × (26/2) × (24/4) = 3 × 13 × 6 = 234 Product of factors of 90 = 90Total factors of 90/2 = 9012/2 = 906 Knowing how to calculate factors of a number is extremely crucial in Maths. The steps to find the factors of a number are given below in a very easy to understand way. An example is taken to make the explanation easier. Find the factors of 16. Step 1: Write the numbers from 1 to 16 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 Step 2: Check now, which number can divide the 16 equally from the above list of numbers. 16/1 = 16 16/2 = 8 16/3 = fraction In the same way, divide 16 by each of these numbers. Step 3: Hence, we get the factors of 16 are 1,2,4,8, and 16 Shortcut Method to Find Factors of a numberLet us see an example, to find the factors of a number using the shortcut method. Consider the number as 40. 40 = 10 × 4 = (5 × 2) × 4 =(5 × 2) × (2 × 2) = (5 × 2) × (2 × 2) Now, the factors of 40 will include all the combinations from 5 × 2 × 2 × 2 and 1 itself (as 1 × 40 = 40). So, the positive factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. It should be noted that there will also be negative factors whose count has to be even. How to Find Factors of Large Numbers?To calculate the factors of large numbers, divide the numbers with the least prime number, i.e. 2. If the number is not divisible by 2, move to the next prime numbers, i.e. 3 and so on until 1 is reached. Below is an example to find the factors of a large number. Example: 1420
In step 3, a prime number is obtained as a product, and so, the process is stopped. The factors will be all the multiples of 1, 2, 2, 5, 71, 355, 710. Now, the positive factors of 1420 will be 1, 2, 4, 5, 10, 20, 71, 142, 284, 355, 710, and 1420. In the same case, if only prime factors are considered, it is called the prime factorization of that number. In this way, it is easy to factor a number and know its factors and prime factors. Prime Factors of a NumberWith the help of prime factorisation method, we can determine the prime factors of a number. Example: Prime factorisation of 144.
Thus, 144 = 2 × 2 × 2 × 2 × 3 × 3. Video Lesson on Prime FactorsList of Factors of NumbersBelow is a list of common numbers with their factors and prime factors. Each of these links will include the process of factoring of any number along with all the factors, including the prime factors of that number. Solved ExamplesQ.1: What are the factors of 8? Solution: By the definition of factors of a number, we know, 1 and the number itself are the two general factors. 8 is an even number and divisible by 2. 8/2 = 4 also, 8/4 = 2 Apart from these numbers, 8 is not evenly divisible from any other number. So, 1, 2, 4 and 8 are the required factors. Q.2: How many factors do 31 has? Solution: Since 31 is a prime number, therefore the factor of 31 are 1 and 31. Therefore, there are only two factors. Q.3: Fill in the blanks: Factors of 9 are 1, ___, 9. Solution: Factors of 9 are 1, 3, 9. Keep visiting BYJU’S to learn more about such Maths concepts in an easy and effective way. Also, register now to get complete assistance for Maths preparation from various video lessons and other study materials. Frequently Asked Questions on Factors of a NumberFactors of a number are the integers that can divide the original number exactly. For example, 8
is the factor of 24, since it can divide into three equal parts. 24/8 = 3 A prime number has only two factors, 1 and number itself. For example, 7 has two factors 1 and 7. The factor of 1 is 1 only. The factors of 5 are 1 and
5 What are the factors of 30?The factors of 30 1, 2, 3, 5, 6, 10, 15 and 30. What are the factors of 12?The factors of 12 are 1,2,3,4,6, and 12 How many positive odd factors does 2022 have?There are 8 positive factors of 2022 and 8 negative factors of 2022. Wht are there negative numbers that can be a factor of 2022?
Why is 2022 a special number?2022 is an untouchable number, because it is not equal to the sum of proper divisors of any number. It is a polite number, since it can be written in 3 ways as a sum of consecutive naturals, for example, 163 + ... + 174. It is an arithmetic number, because the mean of its divisors is an integer number (507).
How many positive integers less than 100 have an odd number of factors?There are 10 numbers between 1 and 100 that have an odd number of factors.
How do you find how many odd factors a number has?So to find the total number of factors, we can add one to each to power in the prime factorization of an integer, then multiply all the (power + 1)s together. For 540, we would have (2 + 1)(3 + 1)(1 + 1) = 24 factors. To find the number of odd factors (which includes 1), we can exclude any power of 2 and do the same.
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