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# Numbers p, q, r and s have 7, 16, 20 and 22 factors. Which of these co

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Joined: 10 Mar 2013
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Numbers p, q, r and s have 7, 16, 20 and 22 factors. Which of these co [#permalink]

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20 Oct 2017, 08:19
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Numbers p, q, r and s have 7, 16, 20 and 22 factors. Which of these could be a perfect cube?

A. p and q
B. q and r
C. r and s
D. p, q and r
E. p, q and s
[Reveal] Spoiler: OA

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Hasan Mahmud

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Joined: 02 Aug 2009
Posts: 5720
Numbers p, q, r and s have 7, 16, 20 and 22 factors. Which of these co [#permalink]

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20 Oct 2017, 08:32
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Mahmud6 wrote:
Numbers p, q, r and s have 7, 16, 20 and 22 factors. Which of these could be a perfect cube?

A. p and q
B. q and r
C. r and s
D. p, q and r
E. p, q and s

Hi...

since the Q is asking COULD, we have to find the possibilities of matching the perfect cube..
7 factors.. 1*7...... so if a number has 6 of a kind that is $$a^6$$, factors = $$1+6=7...a^6= (a^2)^3$$....YES
16 factors.. 1*16...... so if a number has 15 of a kind that is $$a^{15}$$, factors = $$1+15=16...a^{15}= (a^5)^3$$....YES
22 factors.. 1*21...... so if a number has 21 of a kind that is $$a^{21}$$, factors = $$1+21=22...a^{21}= (a^7)^3$$....YES

p, q and s

E
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Numbers p, q, r and s have 7, 16, 20 and 22 factors. Which of these co [#permalink]

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20 Oct 2017, 08:48
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Mahmud6 wrote:
Numbers p, q, r and s have 7, 16, 20 and 22 factors. Which of these could be a perfect cube?

A. p and q
B. q and r
C. r and s
D. p, q and r
E. p, q and s

any cube is of the form $$a^{3k}$$, and hence the number of factors for this number will be $$3k+1$$

This implies that the number of factors when divided by $$3$$ will leave $$1$$ as remainder.

so for our question $$7$$, $$16$$ & $$22$$ will leave a remainder of $$1$$ when divided by $$3$$. Hence $$p$$, $$q$$ & $$s$$ can be a perfect cube

Option E

for the sake of explanation, $$s$$ has $$22$$ factors, so $$s$$ can be of the form $$p_1^{21}$$, this can be written as $$(p_1^7)^3$$ i.e a perfect cube
Numbers p, q, r and s have 7, 16, 20 and 22 factors. Which of these co   [#permalink] 20 Oct 2017, 08:48
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