What is the HCF of 85 in prime factorization method?

The GCF of 85 and 480 is 5.

Steps to find GCF

1. Find the prime factorization of 85
   85 = 5 × 17
2. Find the prime factorization of 480
   480 = 2 × 2 × 2 × 2 × 2 × 3 × 5
3. To find the GCF, multiply all the prime factors common to both numbers:

   Therefore, GCF = 5

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Find hcf of: 170 & 960 255 & 1440 425 & 2400 17 & 96 595 & 3360 170 & 480 85 & 960 255 & 480 85 & 1440 425 & 480 85 & 2400 595 & 480 85 & 3360

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The greatest common factor (GCF) is also known as greatest common divisor (GCD) or highest common factor (HCF).

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gcf, hcf, gcd (85; 221) = ?

The prime factorization of a number: finding the prime numbers that multiply together to make that number.

85 = 5 × 17
85 is not a prime number but a composite one.

221 = 13 × 17
221 is not a prime number but a composite one.

* The natural numbers that are only divisible by 1 and themselves are called prime numbers. A prime number has exactly two factors: 1 and itself.
* A composite number is a natural number that has at least one other factor than 1 and itself.

Multiply all the common prime factors, taken by their smallest powers (exponents).

gcf, hcf, gcd (85; 221) = 17

This algorithm involves the process of dividing numbers and calculating the remainders.

'a' and 'b' are the two natural numbers, 'a' >= 'b'.

Divide 'a' by 'b' and get the remainder of the operation, 'r'.

If 'r' = 0, STOP. 'b' = the gcf (hcf, gcd) of 'a' and 'b'.

Else: Replace ('a' by 'b') and ('b' by 'r'). Return to the step above.

Step 1. Divide the larger number by the smaller one:
221 ÷ 85 = 2 + 51 Step 2. Divide the smaller number by the above operation's remainder:
85 ÷ 51 = 1 + 34 Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
51 ÷ 34 = 1 + 17 Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
34 ÷ 17 = 2 + 0 At this step, the remainder is zero, so we stop: 17 is the number we were looking for - the last non-zero remainder.

This is the greatest (highest) common factor (divisor).

The greatest (highest) common factor (divisor):
gcf, hcf, gcd (85; 221) = 17

Once you've calculated the greatest common factor of the numerator and the denominator of a fraction, it becomes much easier to fully reduce (simplify) the fraction to the lowest terms (the smallest possible numerator and denominator).

Calculate the greatest (highest) common factor (divisor) of numbers, gcd, hcf, gcd:

Method 1: Run the prime factorization of the numbers - then multiply all the common prime factors, taken by their smallest exponents. If there are no common prime factors, then gcf equals 1.

Method 2: The Euclidean Algorithm.

Method 3: The divisibility of the numbers.

• Note 1: The greatest common factor (gcf) is also called the highest common factor (hcf), or the greatest common divisor (gcd).
• Note 2: The Prime Factorization of a number: finding the prime numbers that multiply together to make that number.
• Suppose the number "t" evenly divides the number "a" ( = when evenly dividing the number "a" by "t", the remainder is zero).
• When we look at the prime factorization of "a" and "t", we find that:
• 1) all the prime factors of "t" are also prime factors of "a"
• and
• 2) the exponents of the prime factors of "t" are equal to or smaller than the exponents of the prime factors of "a" (see the * Note below)

• For example, the number 12 is a divisor (a factor) of the number 60:
• 12 = 2 × 2 × 3 = 22 × 3
• 60 = 2 × 2 × 3 × 5 = 22 × 3 × 5
• * Note: 23 = 2 × 2 × 2 = 8. We say that 2 was raised to the power of 3. In this example, 3 is the exponent and 2 is the base. The exponent indicates how many times the base is multiplied by itself. 23 is the power and 8 is the value of the power.

• If the number "t" is a common divisor of the numbers "a" and "b", then:
• 1) "t" only has the prime factors that also intervene in the prime factorization of "a" and "b".
• and
• 2) each prime factor of "t" has the smallest exponents with respect to the prime factors of the numbers "a" and "b".

• For example, the number 12 is the common divisor of the numbers 48 and 360. Below is their prime factorization:
• 12 = 22 × 3
• 48 = 24 × 3
• 360 = 23 × 32 × 5
• You can see that the number 12 has only the prime factors that also occur in the prime factorization of the numbers 48 and 360.
• You can see above that the numbers 48 and 360 have several common factors: 2, 3, 4, 6, 8, 12, 24. Out of these, 24 is the greatest common factor (GCF) of 48 and 360.
• 24 = 2 × 2 × 2 × 3 = 23 × 3
• 48 = 24 × 3
• 360 = 23 × 32 × 5
• 24, the greatest common factor of the numbers 48 and 360, is calculated as the product of all the common prime factors of the two numbers, taken by the smallest exponents (powers).

• If two numbers "a" and "b" have no other common factor than 1, gcf (a, b) = 1, then the numbers "a" and "b" are called coprime numbers (relatively prime, prime to each other).
• If "a" and "b" are not relatively prime numbers, then every common divisor of "a" and "b" is a divisor of the greatest common divisor of "a" and "b".

• Let's have an example on how to calculate the greatest common factor, gcf, of the following numbers:
• 1,260 = 22 × 32
• 3,024 = 24 × 32 × 7
• 5,544 = 23 × 32 × 7 × 11
• gcf (1,260, 3,024, 5,544) = 22 × 32 = 252

• And another example:
• 900 = 22 × 32 × 52
• 270 = 2 × 33 × 5
• 210 = 2 × 3 × 5 × 7
• gcf (900, 270, 210) = 2 × 3 × 5 = 30

• And one more example:
• 90 = 2 × 32 × 5
• 27 = 33
• 22 = 2 × 11
• gcf (90, 27, 22) = 1 - The three numbers have no prime factors in common, they are relatively prime.

What is a prime number? Definition, examples

What is a composite number? Definition, examples

The prime numbers up to 1,000

The prime numbers up to 10,000

The Sieve of Eratosthenes

The Euclidean Algorithm

Completely reduce (simplify) fractions to the lowest terms: Steps and Examples

Factors of 85 are the numbers which when multiplied in pairs give the product as 85. Did you know that the number 85 is the length of the hypotenuse of four Pythagorean triangles? In this lesson, we will calculate the factors of 85, prime factors of 85, and factors of 85 in pairs along with solved examples for a better understanding.

• Factors of 85: 1, 5, 17 and 85
• Prime Factorization of 85: 85 = 5 × 17

What are the Factors of 85?

Factors of a given number are the numbers that divide the given number exactly without any remainder i.e. the remainder is 0. Following are the factors of 85 are 1, 5, 17 and 85. On adding its factors which are proper divisors, we get 1+5+17= 23 < 85 The sum of the proper divisors of 85 is less than 85 So, 85 is known as a deficient number.

Various methods like prime factorization and the division method can be used to calculate the factors of 85. In prime factorization, we express 85 as a product of its prime factors and in the division method, we look for numbers that divide 85 without a remainder.

Important Notes:

• The factors of 85 are 1, 5, 17 and 85
•  As 85 is strictly greater than the sum of its proper divisors, it is called a deficient number
• The smallest factor of a number is 1 and the greatest factor of a number would be the number itself

Hence, the factors of 85 are 1, 5, 17 and 85.

Explore factors using illustrations and interactive examples

• Factors of 75 - The factors of 75 are 1, 3, 5, 15, 25 and 75
• Factors of 54 - The factors of 54 are 1, 2, 3, 6, 9, 18, 27, 54
• Factors of 80 - The factors of 80 are 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
• Factors of 45 - The factors of 168 are 1, 3, 5, 9, 15, 45
• Factors of 175 - The factors of 175 are 1, 5, 7, 25, 35, 175
• Factors of 84 - The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

Factors of 85 by Prime Factorization

Prime 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

The number 85 is divided by the smallest prime number which divides 85 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 85 is an odd number, it will not be divisible by 2 and its multiples. Now, sum of digits of 85 = 8 + 5 = 13 which is not divisible by 3. So, 85 is not divisible by 3 as well. Since the last digit is 5, 85 is divisible by 5

85 ÷ 5=17

Now, 17 is again a prime number and hence can't be further divided. Prime factorization of 85 = 5 × 17

2. Prime Factorization by Factor Tree

The other way of prime factorization as taking 85 as the root, we create branches by dividing it by prime numbers. This method is similar to the above division method. The difference lies in presenting the factorization. The composite numbers will have branches as they are further divisible. We continue making branches till we are left with only prime numbers. Now that we have done the prime factorization of 85, 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 85 in Pairs

The factor pairs of a number are the two numbers which, when multiplied, give the required number.
For example 10 × 9 = 90 and 5 ×18 = 90

So, (10,9) and (5,18) are pair factors of 90.
Considering the number 85, we have:

So, the pair factors of 85 are (1,85) and (5,17).

The product of two negative numbers is positive i.e. (-ve) × (-ve) = (+ve).
So, (-1,-85), (-5,-17)  are also factor pairs of 85.

Challenging Questions:

• Do the numbers 85 and 0.85 have a common factor?
• Is there a natural number n, such that 85n will end with digit 0? How will you justify your answer?

1. Example 1: Ryan's teacher told him that (-5) is one of the factors of 85. Can you help him find the other factor?

   Solution:

   Since, (-5 ) is one the factor of 85, we will divide 85 by (-5) to get the other factor.
   The other factor is 85 ÷ (-5) = -17

   Hence, -17 is the other factor.

2. Example 2: Tom has 85 units of cup sets and he wants to pack them in the cartons such that these units are evenly distributed. There are three sizes of cartons available to him. The first type has a capacity of 5 units, the second, a capacity of 13 units, and the third, a capacity of 17 units. Which type of carton he will choose so that there is no unit left and minimum cartons are required?

   Solution:

   First condition: To find the type of carton such that after packing no unit is left, we need to find the factors of 85. Here 5 and 17 are factors of 85, but 13 is not.

   Second condition: Minimum cartons are required. Let's divide 85 by each of the factors 5 and 17 and decide in which case minimum cartons are required.

85 ÷ 5 = 17  and 85 ÷ 17 = 5

So, when he takes a carton of capacity 17 units he requires only 5 cartons whereas if he takes a carton of capacity 5 units he would require 17 cartons.
Hence, Tom will choose a carton which has a capacity of 17 units.

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FAQs on Factors of 85

What are the factors of 85?

The factors of 85 are 1, 5, 17, and 85.

What are the common factors of 85 and 145?

The factors of 85 are 1, 5, 17, and 85. The factors of 145 are 1, 5, 29, 145.

So, The common factors are 1 and 5.

What are the prime factors of 85?

The  prime factors of 85 are 5 and 17

What are the common factors of 51 and 85?

The factors of 85 are 1, 5, 17, and 85. The factors of 51 are 1, 3, 17, and 51.

So, The common factors are 1 and 17.

What are the common factors of 85 and 115?

The factors of 85 are 1, 5, 17, and 85 The factors of 115 are 1, 5, 23, 115

Hence the common factors are 1, and 5