What is the least number, which is a perfect square but contains 2700 as its factor

The following steps will be useful to find the least number which has to multiplied by the given number to get a perfect square.

1. Decompose the given numbers into its prime factors.

2. Write the prime factors as pairs such that each pair has two same prime factors.

3. Find the prime factor which does not occur in pair. That is the least number to be multiplied by the given number to get a perfect square.

Example 1 :

Find the least number multiplied by 200 to get a perfect square.

Solution :

Decompose 200 into its prime factors.

Prime factors of 200 :

200 = 2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 5

= (2 ⋅ 2) ⋅ 2 ⋅ (5 ⋅ 5)

The prime factor 2 does not occur in pair.

So, '2' is the least number to be multiplied by 200 to get a perfect square.

Justification :

√[2(200)] = √[2(2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 5)]

√400 = √[(2 ⋅ 2)(2 ⋅ 2)(5 ⋅ 5)]

= 2 ⋅ 2 ⋅ 5

= 20

Further,

2(200) = 400 = 202

Example 2 :

Find the least number multiplied by 252 to get a perfect square.

Solution :

Decompose 252 into its prime factors.

Prime factors of 252 :

252 = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 7

= (2 ⋅ 2) ⋅ (3 ⋅ 3) ⋅ 7

The prime factor 7 does not occur in pair.

So, '7' is the least number to be multiplied by 252 to get a perfect square.

Justification :

√[7(252)] = √[7(2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 7)]

√1764 = √[(2 ⋅ 2)(3 ⋅ 3)(7 ⋅ 7)]

= 2 ⋅ 3 ⋅ 7

= 42

Further,

7(252) = 1764 = 422

Example 3 :

Find the least number multiplied by 180 to get a perfect square.

Solution :

Decompose 180 into its prime factors.

Prime factors of 180 :

180 = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5

= (2 ⋅ 2) ⋅ (3 ⋅ 3) ⋅ 5

The prime factor 5 does not occur in pair.

So, '5' is the least number to be multiplied by 180 to get a perfect square.

Justification :

√[5(180)] = √[5(2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5)]

√900 = √[(2 ⋅ 2)(3 ⋅ 3)(5 ⋅ 5)]

= 2 ⋅ 3 ⋅ 5

= 30

Further,

5(180) = 900 = 302

Example 4 :

Find the least number multiplied by 90 to get a perfect square.

Solution :

Decompose 90 into its prime factors.

Prime factors of 90 :

90 = 2 ⋅ 3 ⋅ 3 ⋅ 5

= 2 ⋅ (3 ⋅ 3) ⋅ 5

The prime factors 2 and 5 do not occur in pair.

Product of 2 and 5 :

2 ⋅ 5 = 10

So, '10' is the least number to be multiplied by 90 to get a perfect square.

Justification :

√[10(90)] = √[10(2 ⋅ 3 ⋅ 3 ⋅ 5)]

√900 = √[(2 ⋅ 5)(2 ⋅ 3 ⋅ 3 ⋅ 5)]

= √[(2 ⋅ 2)(3 ⋅ 3)(5 ⋅ 5)]

= 2 ⋅ 3 ⋅ 5

= 30

Further,

10(90) = 900 = 302

Example 5 :

Find the least number multiplied by 120 to get a perfect square.

Solution :

Decompose 120 into its prime factors.

Prime factors of 120 :

120 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5

= (2 ⋅ 2) ⋅ 2 ⋅ 3 ⋅ 5

The prime factors 2, 3 and 5 do not occur in pair.

Product of 2, 3 and 5 :

2 ⋅ 3 ⋅ 5 = 30

So, '30' is the least number to be multiplied by 120 to get a perfect square.

Justification :

√[30(120)] = √[30(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5)]

√3600 = √[(2 ⋅ 3 ⋅ 5)(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5)]

= √[(2 ⋅ 2)(2 ⋅ 2)(3 ⋅ 3)(5 ⋅ 5)]

= 2 ⋅ 2 ⋅ 3 ⋅ 5

= 60

Further,

30(120) = 3600 = 602

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The question is about perfect square factors. A number is given as multiplication of prime factors. We need to find out the number of perfect square factors of that number. Dealing with factors of a number is a vital component in CAT Number Systems: Factors. A range of CAT questions can be asked based on this simple concept of Factors. CAT exam has always tested the idea of Factors from Number systems. The idea of Factors questions forms an integral part of the CAT syllabus.

Question 3: How many factors of 25 * 36 * 52 are perfect squares?

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Any factor of this number should be of the form 2a * 3b * 5c. For the factor to be a perfect square a,b,c have to be even. a can take values 0, 2, 4. b can take values 0, 2, 4, 6 and c can take values 0, 2

Total number of perfect squares = 3 * 4 * 2 = 24

The question is "How many factors of 25 * 36 * 52 are perfect squares?"

Hence the answer is 24.

Choice B is the correct answer.

Factors of 2700 are integers that can be divided evenly into 2700. There are overall 36 factors of 2700 among which 2700 is the biggest factor and 2, 3, 5 are its prime factors. The Prime Factorization of 2700 is 22 × 33 × 52.

• All Factors of 2700: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 27, 30, 36, 45, 50, 54, 60, 75, 90, 100, 108, 135, 150, 180, 225, 270, 300, 450, 540, 675, 900, 1350 and 2700
• Prime Factors of 2700: 2, 3, 5
• Prime Factorization of 2700: 22 × 33 × 52
• Sum of Factors of 2700: 8680

Factors of 2700 are pairs of those numbers whose products result in 2700. These factors are either prime numbers or composite numbers.

How to Find the Factors of 2700?

To find the factors of 2700, we will have to find the list of numbers that would divide 2700 without leaving any remainder.

• 2700/12 = 225; therefore, 12 is a factor of 2700 and 225 is also a factor of 2700.
• 2700/450 = 6; therefore, 450 is a factor of 2700 and 6 is also a factor of 2700.

☛ Also Check:

• Factors of 50 - The factors of 50 are 1, 2, 5, 10, 25, 50
• Factors of 180 - The factors of 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
• Factors of 21 - The factors of 21 are 1, 3, 7, 21
• Factors of 54 - The factors of 54 are 1, 2, 3, 6, 9, 18, 27, 54
• Factors of 10 - The factors of 10 are 1, 2, 5, 10

• 2700 ÷ 2 = 1350
• 1350 ÷ 2 = 675

Further dividing 675 by 2 gives a non-zero remainder. So we stop the process and continue dividing the number 675 by the next smallest prime factor. We stop ultimately if the next prime factor doesn't exist or when we can't divide any further.

So, the prime factorization of 2700 can be written as 22 × 33 × 52 where 2, 3, 5 are prime.

Pair factors of 2700 are the pairs of numbers that when multiplied give the product 2700. The factors of 2700 in pairs are:

• 1 × 2700 = (1, 2700)
• 2 × 1350 = (2, 1350)
• 3 × 900 = (3, 900)
• 4 × 675 = (4, 675)
• 5 × 540 = (5, 540)
• 6 × 450 = (6, 450)
• 9 × 300 = (9, 300)
• 10 × 270 = (10, 270)
• 12 × 225 = (12, 225)
• 15 × 180 = (15, 180)
• 18 × 150 = (18, 150)
• 20 × 135 = (20, 135)
• 25 × 108 = (25, 108)
• 27 × 100 = (27, 100)
• 30 × 90 = (30, 90)
• 36 × 75 = (36, 75)
• 45 × 60 = (45, 60)
• 50 × 54 = (50, 54)

Negative pair factors of 2700 are:

• -1 × -2700 = (-1, -2700)
• -2 × -1350 = (-2, -1350)
• -3 × -900 = (-3, -900)
• -4 × -675 = (-4, -675)
• -5 × -540 = (-5, -540)
• -6 × -450 = (-6, -450)
• -9 × -300 = (-9, -300)
• -10 × -270 = (-10, -270)
• -12 × -225 = (-12, -225)
• -15 × -180 = (-15, -180)
• -18 × -150 = (-18, -150)
• -20 × -135 = (-20, -135)
• -25 × -108 = (-25, -108)
• -27 × -100 = (-27, -100)
• -30 × -90 = (-30, -90)
• -36 × -75 = (-36, -75)
• -45 × -60 = (-45, -60)
• -50 × -54 = (-50, -54)

NOTE: If (a, b) is a pair factor of a number then (b, a) is also a pair factor of that number.

1. Example 1: How many factors are there for 2700?

   Solution:

   The factors of 2700 are too many, therefore if we can find the prime factorization of 2700, then the total number of factors can be calculated using the formula shown below.
   If the prime factorization of the number is ax × by × cz where a, b, c are prime, then the total number of factors can be given by (x + 1)(y + 1)(z + 1).

   Prime Factorization of 2700 = 22 × 33 × 52

   
   Therefore, the total number of factors are (2 + 1) × (3 + 1) × (2 + 1) = 3 × 4 × 3 = 36

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The factors of 2700 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 27, 30, 36, 45, 50, 54, 60, 75, 90, 100, 108, 135, 150, 180, 225, 270, 300, 450, 540, 675, 900, 1350, 2700 and its negative factors are -1, -2, -3, -4, -5, -6, -9, -10, -12, -15, -18, -20, -25, -27, -30, -36, -45, -50, -54, -60, -75, -90, -100, -108, -135, -150, -180, -225, -270, -300, -450, -540, -675, -900, -1350, -2700.

What is the Sum of the Factors of 2700?

Sum of all factors of 2700 = (22 + 1 - 1)/(2 - 1) × (33 + 1 - 1)/(3 - 1) × (52 + 1 - 1)/(5 - 1) = 8680

What numbers are the Pair Factors of 2700?

The pair factors of 2700 are (1, 2700), (2, 1350), (3, 900), (4, 675), (5, 540), (6, 450), (9, 300), (10, 270), (12, 225), (15, 180), (18, 150), (20, 135), (25, 108), (27, 100), (30, 90), (36, 75), (45, 60), (50, 54).

What is the Greatest Common Factor of 2700 and 1083?

The factors of 2700 and 1083 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 27, 30, 36, 45, 50, 54, 60, 75, 90, 100, 108, 135, 150, 180, 225, 270, 300, 450, 540, 675, 900, 1350, 2700 and 1, 3, 19, 57, 361, 1083 respectively. Common factors of 2700 and 1083 are [1, 3].

Hence, the Greatest Common Factor of 2700 and 1083 is 3.

How Many Factors of 2700 are also Factors of 943?

Since, the factors of 2700 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 27, 30, 36, 45, 50, 54, 60, 75, 90, 100, 108, 135, 150, 180, 225, 270, 300, 450, 540, 675, 900, 1350, 2700 and factors of 943 are 1, 23, 41, 943. Hence, 2700 and 943 have only one common factor which is 1. Therefore, 2700 and 943 are co-prime.