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Factors of 84 are integers that can be divided evenly into 84. It has total 12 factors of which 84 is the biggest factor and the prime factors of 84 are 2, 3 and 7. The Prime Factorization of 84 is 22 × 3 × 7.
What are the Factors of 84?Factors of a given number are the numbers which divide the given number exactly without any remainder. 84 is strictly smaller than the sum of its factors (except 84). 1 + 2 + 3 + 4 + 6 + 7 + 12 + 14 + 21 + 28 + 42 = 140 So, 84 is known as an abundant number. How to Calculate the Factors of 84?Various methods such as prime factorization and the division method can be used to calculate the factors of 84. In prime factorization, we express 84 as a product of its prime factors and in the division method, we see what numbers divide 84 exactly without a remainder. Hence, the factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84. Explore factors using illustrations and interactive examples.
Factors of 84 by Prime FactorizationPrime factorization means expressing a number in terms of the product of its prime factors. We can do use the division method or factor tree to do this. 1. Prime Factorization by Division methodThe number 84 is divided by the smallest prime number which divides 84 exactly, i.e. it leaves a remainder 0. The quotient is then divided by the smallest or second smallest prime number and the process continues till the quotient becomes indivisible. Since 84 is even, it will be divisible by 2. 84 ÷ 2 = 42 42 is again even, let's again divide by 2 42 ÷ 2 = 21 Now, 21 is an odd number. So it won't be divisible by 2. Let's check the next prime number i.e. 3 21 ÷ 3 =7 7 is a prime number. So, it's not further divisible. Prime factorization of 84 = 2 × 2 × 3 × 7 2. Prime Factorization by Factor TreeThe other way of prime factorization as taking 84 as the root, we create branches by dividing it by prime numbers. This method is similar to above division method. The difference lies in presenting the factorization.  The numbers inside the circles are the prime factors of 84 Now that we have done the prime factorization of our number, we can multiply them and get the other factors. Can you try and find out if all the factors are covered or not? And as you might have already guessed it, for prime numbers, there are no other factors. Factors of 84 in PairsThe factor pairs of a number are the two numbers which, when multiplied, give the required number.
So, the factor pairs of 84 are : (1, 84), (2, 42), (3, 28), (4, 21), (6, 14), (7, 12) The product of two negative numbers also results in a positive number i.e. (-ve) × (-ve) = (+ve). So, (-1, -84), (-2, -42), (-3, -28), (-4, -21), (-6, -14), (-7, -12). are also factor pairs of 84. Our focus in this article will be on the positive factors. Important Notes:
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The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 and its negative factors are -1, -2, -3, -4, -6, -7, -12, -14, -21, -28, -42, -84. What are the Prime Factors of 84?The prime factors of 84 are 2, 3, 7. How Many Factors of 84 are also common to the Factors of 44?Since, the factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 and the factors of 44 are 1, 2, 4, 11, 22, 44. What is the Sum of all the Factors of 84?Sum of all factors of 84 = (22 + 1 - 1)/(2 - 1) × (31 + 1 - 1)/(3 - 1) × (71 + 1 - 1)/(7 - 1) = 224 What is the Greatest Common Factor of 84 and 72?The factors of 84 and 72 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 and 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 respectively. Common factors of 84 and 72 are [1, 2, 3, 4, 6, 12]. Hence, the Greatest Common Factor of 84 and 72 is 12. Answer Hint: Assume the two numbers in such a way that their sum is 84. Then, form an expression from the given condition and apply the rule of derivatives to find the maximum value of the expression and hence the required numbers. Complete step-by-step answer: Let the required numbers be x and 84-x.Let $P=x(84-x)^2$We have to find the numbers such that the value of P is maximum.Differentiating P w.r.t. x, we get,$\dfrac{dP}{dx}=\dfrac{d}{dx}[x(84-x)^2]$Applying product rule of differentiation, we get,$\dfrac{dP}{dx}=(84-x)^2+2x(84-x)(-1)$Simplifying, we get,$\dfrac{dP}{dx}=(84-x)^2-2x(84-x)$Again, differentiating it w.r.t. x, we get,$\dfrac{d^2P}{dx^2}=\dfrac{d}{dx}[(84-x)^2-2x(84-x)]$Simplifying, we get,$\dfrac{d^2P}{dx^2}=\dfrac{d}{dx}[(84-x)^2]-\dfrac{d}{dx}[2x(84-x)]$$\implies \dfrac{d^2P}{dx^2}=2(84-x)(-1)-[2x(-1)+(84-x)(2)]$$\implies \dfrac{d^2P}{dx^2}=-2(84-x)-[-2x+2(84-x)]$$\implies \dfrac{d^2P}{dx^2}=-2(84-x)+2x-2(84-x)$$\implies \dfrac{d^2P}{dx^2}=-4(84-x)+2x$Now, to find maxima or minima, we put $\dfrac{dP}{dx}=0$.Thus, $\dfrac{dP}{dx}=(84-x)^2-2x(84-x)=0$ $\implies (84-x)^2-2x(84-x)=0$Rearranging the terms, we get,$(84-x)^2=2x(84-x)$$\implies (84-x)=2x$Again, rearranging the terms, we get,$2x+x=84$$\implies 3x=84$$\implies x =28$Now, we find the value of $\dfrac{d^2P}{dx^2}$ at x =28. If $\dfrac{d^2P}{dx^2}$> 0 at x = 28, then x = 28 is a point of minima and if $\dfrac{d^2P}{dx^2}$ < 0 at x = 28, then x = 28 is a point of maxima.Now, $\dfrac{d^2P}{dx^2}=-4(84-x)+2x$At x = 28,$\dfrac{d^2P}{dx^2}=-4(84-28)+2(28)=-224+56=-168<0$As $\dfrac{d^2P}{dx^2}$< 0, x = 28 is a point of maxima.Now, if x =28 then, 84-x = 56.Hence, the required numbers are 28 and 56. Note: This type of question is an application of derivatives. In this type of question where we have to find the maximum or minimum value of a function, we use a second derivative test. Firstly, we find the first derivative of the function and by equating it to zero, we get the critical points. Putting the value of the critical point in the second derivative, if the resulting value is less than zero, the critical point is a point of maximum and vice-versa. |
Divide 84 into two parts such that the product of one part and the square of the other is maximum
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