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10 Feb 2020, 23:01
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
24% (02:18) correct
76%
(02:31)
wrong
based on 204
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How many factors does positive integer z have?
(1) z/5 and z/7 are integers and the greatest integer that divides them both is 8.
(2) The smallest integer that is divisible by both z and 14 is 280.
(1) z/5 and z/7 are integers and the greatest integer that divides them both is 8.
(2) The smallest integer that is divisible by both z and 14 is 280.
18 Feb 2020, 04:10
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Raxit85
Solution
Step 1: Analyse Question Stem
- • z is a positive integer.
• We need to find the number of factors of z.
- o To find the number of factors we need to know the prime factorization of z.
o So, let us assume that \(z = 2^a*3^b*5^c*7^d*11^e…\) and so on.
Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE
Statement 1: \(z/5\) and \(z/7 \)are integers and the greatest integer that divides them both is 8.
- • \(\frac{z}{5} = 2^a*3^b*5^{c-1} *7^d*11^e… \)and so on.
• \(\frac{z}{7} = 2^a*3^b*5^c*7^{d-1}*11^e…\) and so on.
• Now, H.C.F of \(\frac{z}{5}\) and \(\frac{z}{7}\)\( = 2^a*3^b*5^{c-1} *7^{d-1}*11^e… \)and so on. \(= 2^3\)
- o This means :
- o \(a =3\)
o \(b = 0\),
o \(c -1 = 0\) or \(c =1\)
o \(d-1 = 0 \)or \(d = 1 \)
o e and powers of all further primes are zero.
- o Hence, the total number of factors of z = (3+1)*(1+1)*(1+1)
Statement 2: The smallest integer that is divisible by both z and 14 is 280.
- • According to this statement, LCM of \(z\) and \(14 = 280 = 2^3*5*7\)
• Now, \(14 = 2*7\)
• So, z can be \(2^3*5*7\) or \(2^3*5\)
- o If \(z = 2^3*5*7\), the total number of factors \(= (3+1)*(1+1)*(1+1)\) and
o If \(z = 2^3*5\), the total number of factors \(= (3+1)*(1+1)\)
Thus, the correct answer is Option A.
General Discussion
24 Sep 2020, 07:57
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Hi Bunuel
I wish to practice this concept because I am not able to solve these questions at ease.
Can you suggest some questions or a tag under which I can get these questions
TIA!
I wish to practice this concept because I am not able to solve these questions at ease.
Can you suggest some questions or a tag under which I can get these questions
TIA!
