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23 Sep 2022, 03:21
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C
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Difficulty:
15%
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Question Stats:
76% (01:02) correct
24%
(00:45)
wrong
based on 41
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If p is an integer, then p is divisible by how many positive integers?
(1) The only prime factors of p are 2, 3, and 5.
(2) p < 50
(1) The only prime factors of p are 2, 3, and 5.
(2) p < 50
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23 Sep 2022, 11:03
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carcass
Given: p = Integer
To determine: Number of factors of p
Pre-think Information: Number of factors of a number of format \(a^x b^y c^z\) (Where a,b,c are prime numbers, and x,y,z are integer powers) = (x+1)(y+1)(z+1)
(1) The only prime factors of p are 2, 3, and 5.
\(p = 2^a 3^b 5^c\)
Number of factors = \((a+1)(b+1)(c+1)\)
Now, we are given that p has only three prime factors but powers are not mentioned, number of factors will depend on the powers
NOT SUFFICIENT
(2) p < 50
Different numbers below 50 will have different number of factors
NOT SUFFICIENT
(1) + (2) Combined:
\(p = 2^a 3^b 5^c\)
Smallest Possible Number = \(2^1 3^1 5^1 = 30\)
Now the next possible number in this list can be if we raise 2 to the power of 2 so the resultant number will be = \(2^2 3^1 5^1 = 60\)
But since p < 50, so this is not possible
So, only one number is possible (p = 30)
SUFFICIENT
Answer - C
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