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For positive integers a, b, and c, where a<b<c, the product abc has ex

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Bunuel
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For positive integers a, b, and c, where a<b<c, the product abc has ex [#permalink] For positive integers a, b, and c, where a<b<c, the product abc has ex   22 Feb 2017, 06:26
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For positive integers a, b, and c, where a<b<c, the product abc has exactly three prime factors. What is the maximum number of prime factors of a?

A. 0
B. 1
C. 2
D. 3
E. 4
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D
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Spartan85
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Re: For positive integers a, b, and c, where a<b<c, the product abc has ex [#permalink] For positive integers a, b, and c, where a<b<c, the product abc has ex   22 Feb 2017, 06:48
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Answer : D

If product a*b*c has exactly 3 prime factors,

a can have maximum 3 prime factors.

eg.
a= 2^1 * 3^1 * 5^1
b = 2^2 * 3^2 * 5^2
c = 2^3 * 3^3 * 5^3

Now a * b* c = 2^6 * 3^6 * 5^6
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rohit8865
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For positive integers a, b, and c, where a<b<c, the product abc has ex [#permalink] For positive integers a, b, and c, where a<b<c, the product abc has ex   22 Feb 2017, 08:13
Bunuel wrote:
For positive integers a, b, and c, where a<b<c, the product abc has exactly three prime factors. What is the maximum number of prime factors of a?

A. 0
B. 1
C. 2
D. 3
E. 4


for maximum no. of prime factors of a , it should be multiple of three prime nos.(since abc has only three prime factors)
and b &c could have any number of those three prime factors in a.

say if a =5^x * 7^y * 2^z
then b = 5^x * 7^y or any such prime combinations
& c= 5^x or 7^y or any combination as per condition a<b<c

thus a have max. of 3 prime factors

Ans D
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Re: For positive integers a, b, and c, where a<b<c, the product abc has ex [#permalink] For positive integers a, b, and c, where a<b<c, the product abc has ex   22 Feb 2017, 12:25
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Option D

Number of prime factors of a will be maximum when a has same prime factors as the product a*b*c.
So, the maximum number of prime factors a can have = number of prime factors of product a*b*c = 3.
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Re: For positive integers a, b, and c, where a<b<c, the product abc has ex [#permalink] For positive integers a, b, and c, where a<b<c, the product abc has ex   03 Apr 2021, 19:16
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Nice trap question.

The product result has exactly three prime factors.

The question doesn’t say that one prime factor comes from each number, the 3 prime factors can be present in all three value of A < B < C

A = (2) (3) (5)

B = (2)^2 (3)^2 (5)

C = (2)^3 (3)^3 (5)

Result of multiplication:

ABC = (2)^6 * (3)^6 * (5)^3

Condition is met- Result of ABC is divisible by exactly 3 prime factors 2, 3, and 5

And A can have all 3 prime factors

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Re: For positive integers a, b, and c, where a<b<c, the product abc has ex [#permalink] For positive integers a, b, and c, where a<b<c, the product abc has ex   05 Jul 2024, 21:46
 
Spartan85 wrote:
Answer : D

If product a*b*c has exactly 3 prime factors,

a can have maximum 3 prime factors.

eg.
a= 2^1 * 3^1 * 5^1
b = 2^2 * 3^2 * 5^2
c = 2^3 * 3^3 * 5^3

Now a * b* c = 2^6 * 3^6 * 5^6

­The question doesn't say that it has 3 distinct prime factors. It simply says 3 prime factors, and usually, unless the question stem explicitly says 'distinct', we take TOTAL number of prime numbers into account. So there are 6 + 6 + 6 = 18 prime factors in your expression. 
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Re: For positive integers a, b, and c, where a<b<c, the product abc has ex [#permalink] For positive integers a, b, and c, where a<b<c, the product abc has ex   06 Jul 2024, 01:07
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ZIX wrote:
Spartan85 wrote:
Answer : D

If product a*b*c has exactly 3 prime factors,

a can have maximum 3 prime factors.

eg.
a= 2^1 * 3^1 * 5^1
b = 2^2 * 3^2 * 5^2
c = 2^3 * 3^3 * 5^3

Now a * b* c = 2^6 * 3^6 * 5^6

­The question doesn't say that it has 3 distinct prime factors. It simply says 3 prime factors, and usually, unless the question stem explicitly says 'distinct', we take TOTAL number of prime numbers into account. So there are 6 + 6 + 6 = 18 prime factors in your expression. 


­Not so. Three prime factors means three distinct prime factors. 
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Re: For positive integers a, b, and c, where a<b<c, the product abc has ex [#permalink]
06 Jul 2024, 01:07
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