When it comes to mathematics, one of the key concepts is understanding the divisors or factors of a number. These are the numbers that divide the given number completely without leaving a remainder. In this guide, we will delve deep into the concepts of divisors, how to calculate them, and explore important theorems, such as the number of divisors, sum of divisors, and even and odd factors.
1. What Are Divisors and Factors?
A divisor of a number NNN is any number that divides NNN completely. For example, the number 24 has divisors like 1, 2, 3, 4, 6, 8, 12, and 24. These are all the numbers that divide 24 exactly, with no remainder.Example:
For N=24N = 24N=24, the divisors are: 1,2,3,4,6,8,12,241, 2, 3, 4, 6, 8, 12, 241,2,3,4,6,8,12,242. Number of Divisors
The number of divisors of a number can be calculated using its prime factorization. If a number NNN is expressed as:N=xa⋅yb⋅zc⋯N = x^a \cdot y^b \cdot z^c \cdotsN=xa⋅yb⋅zc⋯Then, the number of divisors of NNN is calculated using the formula:Number of divisors=(a+1)(b+1)(c+1)⋯\text{Number of divisors} = (a + 1)(b + 1)(c + 1) \cdotsNumber of divisors=(a+1)(b+1)(c+1)⋯This works because for each prime factor, the exponents can vary from 0 to the exponent of that prime in the factorization.Example:
For N=7200=25⋅32⋅52N = 7200 = 2^5 \cdot 3^2 \cdot 5^2N=7200=25⋅32⋅52, the number of divisors is:(5+1)⋅(2+1)⋅(2+1)=6⋅3⋅3=54(5 + 1) \cdot (2 + 1) \cdot (2 + 1) = 6 \cdot 3 \cdot 3 = 54(5+1)⋅(2+1)⋅(2+1)=6⋅3⋅3=54Thus, 7200 has 54 divisors.3. Even and Odd Divisors
We can also break down the divisors of a number into even and odd divisors.Formula for Even Divisors:
If N=2a⋅xb⋅yc⋯N = 2^a \cdot x^b \cdot y^c \cdotsN=2a⋅xb⋅yc⋯, then the number of even divisors is:Number of even divisors=a⋅(b+1)⋅(c+1)⋯\text{Number of even divisors} = a \cdot (b + 1) \cdot (c + 1) \cdotsNumber of even divisors=a⋅(b+1)⋅(c+1)⋯Formula for Odd Divisors:
The odd divisors are the divisors of the number obtained by excluding the power of 2 in the prime factorization of NNN. Hence:Number of odd divisors=(b+1)⋅(c+1)⋯\text{Number of odd divisors} = (b + 1) \cdot (c + 1) \cdotsNumber of odd divisors=(b+1)⋅(c+1)⋯4. Sum of Divisors
The sum of divisors of NNN, where N=xa⋅yb⋅zc⋯N = x^a \cdot y^b \cdot z^c \cdotsN=xa⋅yb⋅zc⋯, is given by:Sum of divisors=(xa+1−1)(x−1)⋅(yb+1−1)(y−1)⋯\text{Sum of divisors} = \frac{(x^{a+1} – 1)}{(x – 1)} \cdot \frac{(y^{b+1} – 1)}{(y – 1)} \cdotsSum of divisors=(x−1)(xa+1−1)⋅(y−1)(yb+1−1)⋯Example:
For N=120=23⋅31⋅51N = 120 = 2^3 \cdot 3^1 \cdot 5^1N=120=23⋅31⋅51, the sum of divisors is:(24−1)(2−1)⋅(32−1)(3−1)⋅(52−1)(5−1)=360\frac{(2^{4} – 1)}{(2 – 1)} \cdot \frac{(3^{2} – 1)}{(3 – 1)} \cdot \frac{(5^{2} – 1)}{(5 – 1)} = 360(2−1)(24−1)⋅(3−1)(32−1)⋅(5−1)(52−1)=3605. Product of Divisors
The product of divisors of NNN is given by:Product of divisors=Nnumber of divisors2\text{Product of divisors} = N^{\frac{\text{number of divisors}}{2}}Product of divisors=N2number of divisorsThis formula works because the divisors come in pairs that multiply to NNN.Example:
For N=120N = 120N=120, the number of divisors is 16, so the product of divisors is:120162=1208120^{\frac{16}{2}} = 120^8120216=12086. Perfect Squares and Divisors
Perfect squares have an odd number of divisors because every divisor pairs up with another, except the square root of the number.Example:
For N=36=22⋅32N = 36 = 2^2 \cdot 3^2N=36=22⋅32, the divisors are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Notice that 6 pairs with itself (since 6×6=366 \times 6 = 366×6=36).7. Totient Function (Euler’s Phi Function)
The totient function, denoted ϕ(N)\phi(N)ϕ(N), calculates the number of integers less than NNN that are coprime to NNN. For N=xa⋅yb⋅zc⋯N = x^a \cdot y^b \cdot z^c \cdotsN=xa⋅yb⋅zc⋯, the formula is:ϕ(N)=N(1−1x)(1−1y)(1−1z)⋯\phi(N) = N \left( 1 – \frac{1}{x} \right) \left( 1 – \frac{1}{y} \right) \left( 1 – \frac{1}{z} \right) \cdotsϕ(N)=N(1−x1)(1−y1)(1−z1)⋯8. Key Examples to Practice
Example 1: How Many Divisors Does 7200 Have?
7200=25⋅32⋅527200 = 2^5 \cdot 3^2 \cdot 5^27200=25⋅32⋅52The number of divisors is:(5+1)(2+1)(2+1)=54(5+1)(2+1)(2+1) = 54(5+1)(2+1)(2+1)=54Example 2: Find the Odd and Even Divisors of 84
84=22⋅31⋅7184 = 2^2 \cdot 3^1 \cdot 7^184=22⋅31⋅71The total number of divisors is:(2+1)(1+1)(1+1)=12(2+1)(1+1)(1+1) = 12(2+1)(1+1)(1+1)=12The odd divisors are 1,3,7,211, 3, 7, 211,3,7,21, so there are 4 odd divisors. Hence, the number of even divisors is:12−4=812 – 4 = 812−4=89. Conclusion: How to Master Divisors and Factors
Understanding divisors, prime factorizations, and the formulas for calculating divisors, sums, and products is crucial for solving mathematical problems efficiently, especially in competitive exams and campus placements. By practicing the above formulas and examples, students can sharpen their understanding and excel in their exams.Visual Suggestions for Each Section:
- Prime Factorization Visual: A diagram showing the prime factorization of numbers like 7200 and 84.
- Formula Graphics: Create simple graphics displaying the divisor formulas for quick reference.
- Interactive Factor Tree: Visualize a factor tree for a composite number to help students visualize the process of finding divisors.
- Perfect Square Visualization: Show how divisors of perfect squares form pairs, emphasizing the unique case where the square root is a single unpaired divisor.