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The integer 120 has many factorizations. For example, 120 = (2)(60), 1
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parkhydel
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The integer 120 has many factorizations. For example, 120 = (2)(60), 1 [#permalink] New post  27 Apr 2020, 14:52
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The integer 120 has many factorizations. For example, \(120 = (2)(60)\), \(120 = (3)(4)(10)\), and \(120 = (–1)(–3)(4)(10)\). In how many of the factorizations of 120 are the factors consecutive integers in ascending order?

A. 2
B. 3
C. 4
D. 5
E. 6


PS62451.02
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GaurSaini
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Re: The integer 120 has many factorizations. For example, 120 = (2)(60), 1 [#permalink] New post  01 May 2020, 20:35
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let's do prime factorization of 120 so that we can look into the possible solutions.

120 = 2^3*3*5

Looking at the factorization, we have 2,3,5 and power of 2.

Hence we can deduce that 2,3,4,5 are factors of 120.

Hence,
1. 1*2*3*4*5
2. 2*3*4*5
3. -5*-4*-3*-2
4. 4*5*6
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nick1816
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The integer 120 has many factorizations. For example, 120 = (2)(60), 1 [#permalink] New post  27 Apr 2020, 15:18
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Minimum product of 6 consecutive positive numbers = 6!=720
So we can't write 120 as the product of 6 or more consecutive positive or negative integers

1. Product of minimum 5 positive integers = 5! = 120

120 = 1*2*3*4*5

2. \(3^4 < 120<4^4\)
Hence if 120 can be written as the product of 4 consecutive integers, then 3 and 4 must be included.

\(\frac{120}{3*4} = 10 =2*5\)

120 = 2*3*4*5 (since number of factors are even, we can change the signs of each factor and make another combination)

3. 120= -5*-4*-3*-2

4. \(4^3<120<5^3\)
Hence if 120 can be written as the product of 3 consecutive integers, then 4 and 5 must be included.

120/4*5 = 6

120= 4*5*6

\(10^2<120<11^2\)

10*11 is not equal to 120



C




parkhydel wrote:
The integer 120 has many factorizations. For example, \(120 = (2)(60)\), \(120 = (3)(4)(10)\), and \(120 = (–1)(–3)(4)(10)\). In how many of the factorizations of 120 are the factors consecutive integers in ascending order?

A. 2
B. 3
C. 4
D. 5
E. 6


PS62451.02
avatar
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lacktutor
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Joined: 25 Jul 2018
Posts: 733
Re: The integer 120 has many factorizations. For example, 120 = (2)(60), 1 [#permalink] New post  27 Apr 2020, 15:36
nick1816 wrote:
120 = 1*2*3*4*5
120 = 2*3*4*5 (since number of factors are even, we can change the signs of each factor and make another combination)
120= -5*-4*-3*-2
120= 4*5*6

D



parkhydel wrote:
The integer 120 has many factorizations. For example, \(120 = (2)(60)\), \(120 = (3)(4)(10)\), and \(120 = (–1)(–3)(4)(10)\). In how many of the factorizations of 120 are the factors consecutive integers in ascending order?

A. 2
B. 3
C. 4
D. 5
E. 6


PS62451.02


Hi, nick1816
Are there 5 or 4 possible combinations in total?
You provided 4 possible combinations, but chose option D.

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nick1816
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Joined: 19 Oct 2018
Posts: 2248
Location: India
Re: The integer 120 has many factorizations. For example, 120 = (2)(60), 1 [#permalink] New post  27 Apr 2020, 15:38
That was a typo. Edited!

Thanks

lacktutor wrote:
nick1816 wrote:
120 = 1*2*3*4*5
120 = 2*3*4*5 (since number of factors are even, we can change the signs of each factor and make another combination)
120= -5*-4*-3*-2
120= 4*5*6

D



parkhydel wrote:
The integer 120 has many factorizations. For example, \(120 = (2)(60)\), \(120 = (3)(4)(10)\), and \(120 = (–1)(–3)(4)(10)\). In how many of the factorizations of 120 are the factors consecutive integers in ascending order?

A. 2
B. 3
C. 4
D. 5
E. 6


PS62451.02


Hi, nick1816
Are there 5 or 4 possible combinations in total?
You provided 4 possible combinations, but chose option D.

Posted from my mobile device
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Rajat8
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Joined: 06 Oct 2019
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Re: The integer 120 has many factorizations. For example, 120 = (2)(60), 1 [#permalink] New post  11 Jan 2021, 18:48
plaverbach wrote:
Why not -4*-5*6?


Hey plaverbach , that's cause in -5, -4, 6 combination, -4 and 6 are not consecutive integers.

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nisthagupta28
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Joined: 28 Mar 2019
Posts: 22
Re: The integer 120 has many factorizations. For example, 120 = (2)(60), 1 [#permalink] New post  17 Feb 2021, 20:23
GaurSaini wrote:
let's do prime factorization of 120 so that we can look into the possible solutions.

120 = 2^3*3*5

Looking at the factorization, we have 2,3,5 and power of 2.

Hence we can deduce that 2,3,4,5 are factors of 120.

Hence,
1. 1*2*3*4*5
2. 2*3*4*5
3. -5*-4*-3*-2
4. 4*5*6




how are 1 and 2 different in the end product?
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nisthagupta28
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Joined: 28 Mar 2019
Posts: 22
Re: The integer 120 has many factorizations. For example, 120 = (2)(60), 1 [#permalink] New post  17 Feb 2021, 20:26
nick1816 wrote:
Minimum product of 6 consecutive positive numbers = 6!=720
So we can't write 120 as the product of 6 or more consecutive positive or negative integers

1. Product of minimum 5 positive integers = 5! = 120

120 = 1*2*3*4*5

2. \(3^4 < 120<4^4\)
Hence if 120 can be written as the product of 4 consecutive integers, then 3 and 4 must be included.

\(\frac{120}{3*4} = 10 =2*5\)

120 = 2*3*4*5 (since number of factors are even, we can change the signs of each factor and make another combination)

3. 120= -5*-4*-3*-2

4. \(4^3<120<5^3\)
Hence if 120 can be written as the product of 3 consecutive integers, then 4 and 5 must be included.

120/4*5 = 6

120= 4*5*6

\(10^2<120<11^2\)

10*11 is not equal to 120



C




parkhydel wrote:
The integer 120 has many factorizations. For example, \(120 = (2)(60)\), \(120 = (3)(4)(10)\), and \(120 = (–1)(–3)(4)(10)\). In how many of the factorizations of 120 are the factors consecutive integers in ascending order?

A. 2
B. 3
C. 4
D. 5
E. 6

PS62451.02



then 3 and 4 must be included?? Can you explain this.
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Gknight5603
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Joined: 25 Oct 2019
Posts: 99
Re: The integer 120 has many factorizations. For example, 120 = (2)(60), 1 [#permalink] New post  17 Feb 2021, 20:39
GaurSaini wrote:
let's do prime factorization of 120 so that we can look into the possible solutions.

120 = 2^3*3*5

Looking at the factorization, we have 2,3,5 and power of 2.

Hence we can deduce that 2,3,4,5 are factors of 120.

Hence,
1. 1*2*3*4*5
2. 2*3*4*5
3. -5*-4*-3*-2
4. 4*5*6


why are we not including
-5*-4*-3*-2*-1

Posted from my mobile device
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GaurSaini
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Joined: 06 Feb 2020
Posts: 28
Re: The integer 120 has many factorizations. For example, 120 = (2)(60), 1 [#permalink] New post  18 Feb 2021, 00:43
Gknight5603 wrote:
GaurSaini wrote:
let's do prime factorization of 120 so that we can look into the possible solutions.

120 = 2^3*3*5

Looking at the factorization, we have 2,3,5 and power of 2.

Hence we can deduce that 2,3,4,5 are factors of 120.

Hence,
1. 1*2*3*4*5
2. 2*3*4*5
3. -5*-4*-3*-2
4. 4*5*6


why are we not including
-5*-4*-3*-2*-1

Posted from my mobile device


we are not including -5*-4*-3*-2*-1 because it will lead to -120 not 120.

Hope it helps
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