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If the least common multiple of a positive integer x ,4^3 and 6^5 is 6

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If the least common multiple of a positive integer x ,4^3 and 6^5 is 6  [#permalink]

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04 Dec 2014, 01:46
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Difficulty:

85% (hard)

Question Stats:

48% (01:59) correct 52% (02:19) wrong based on 183 sessions

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If the least common multiple of a positive integer x ,4^3 and 6^5 is 6^6. Then x can take how many values?

A 1
B 6
C 7
D 30
E 36
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Joined: 02 Sep 2009
Posts: 50711
Re: If the least common multiple of a positive integer x ,4^3 and 6^5 is 6  [#permalink]

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04 Dec 2014, 03:17
3
2
anceer wrote:
If the least common multiple of a positive integer x ,4^3 and 6^5 is 6^6. Then x can take how many values?

A 1
B 6
C 7
D 30
E 36

If the least common multiple of a positive integer x, 4^3 and 6^5 is 6^6. Then x can take how many values?

A. 1
B. 6
C. 7
D. 30
E. 36

We are given that $$6^6=2^{6}*3^{6}$$ is the least common multiple of the following three numbers:

x;
$$4^3=2^6$$;
$$6^5 = 2^{5}*3^5$$;

First notice that $$x$$ cannot have any other primes other than 2 or/and 3, because LCM contains only these primes.

Now, since the power of 3 in LCM is higher than the powers of 3 in either the second number or in the third, than $$x$$ must have $$3^{6}$$ as its multiple (else how $$3^{6}$$ would appear in LCM?).

Next, $$x$$ can have 2 as its prime in ANY power ranging from 0 to 6, inclusive (it cannot have higher power of 2 since LCM limits the power of 2 to 6).

Thus, $$x$$ could take total of 7 values.

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Re: If the least common multiple of a positive integer x ,4^3 and 6^5 is 6  [#permalink]

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04 Dec 2014, 02:53
2
anceer wrote:
If the least common multiple of a positive integer x ,4^3 and 6^5 is 6^6. Then x can take how many values?

A 1
B 6
C 7
D 30
E 36

$$4^3=2^6$$, $$6^5= 2^5.3^5$$

now if the power of 3 in x is less than 6, then l.c.m will have $$3^5$$in it.
if the power of 3 in x is more than 6, then l.c.m will have a term more than $$3^6$$.

thus x must have $$3^6$$ in it, and x cannot have any prime number other than 2 and 3. why ?? because in that case l.c.m won't be $$6^6$$.
Also, maximum power of 2 that x can contain is 6. thus power of 2 in x can vary from 2^0 to 2^6. this gives us 7 possible cases. which are $$2^0.3^6,2^1.3^6,2^2.3^6,2^3.3^6,2^4,3^6,2^5.3^6,2^6.3^6$$

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Re: If the least common multiple of a positive integer x ,4^3 and 6^5 is 6  [#permalink]

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15 Oct 2017, 08:13
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Re: If the least common multiple of a positive integer x ,4^3 and 6^5 is 6 &nbs [#permalink] 15 Oct 2017, 08:13
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