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27 Apr 2020, 15:52
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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
A. 2
B. 3
C. 4
D. 5
E. 6
PS62451.02
01 May 2020, 21:35
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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
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
27 Apr 2020, 16:18
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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
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
