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# If k and m are integers and k = m*(m+4)*(m+5), k must be divisible by

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Intern
Joined: 16 Mar 2017
Posts: 13
If k and m are integers and k = m*(m+4)*(m+5), k must be divisible by  [#permalink]

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19 Sep 2017, 07:23
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25% (medium)

Question Stats:

73% (01:28) correct 27% (01:40) wrong based on 200 sessions

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If k and m are integers and k = m*(m+4)*(m+5), k must be divisible by which of following(s)?

I. 3
II. 6
III. 8

A. I
B. I and II
C. II and III
D. II
E. III
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Joined: 25 Feb 2013
Posts: 1167
Location: India
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If k and m are integers and k = m*(m+4)*(m+5), k must be divisible by  [#permalink]

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19 Sep 2017, 09:34
2
3
petrified17 wrote:
If k and m are integers and k = m*(m+4)*(m+5), k must be divisible by which of following(s)?

I. 3
II. 6
III. 8

A. I
B. I and II
C. II and III
D. II
E. III

Mathematical approach

$$k=m*(m+4)*(m+5) = (m+3-3)*(m+4)*(m+5)$$

or $$k = (m+3)*(m+4)*(m+5)-3(m+4)*(m+5)$$

Now, $$(m+3)$$, $$(m+4)$$ & $$(m+5)$$ are three consecutive integers, hence MUST be divisible by $$3$$ & $$6$$

$$-3(m+4)*(m+5)$$, has $$3$$ as a factor and $$(m+4)$$ & $$(m+5)$$ are consecutive integers, hence either of the two will be a multiple of $$2$$. Hence the product MUST be multiple of $$3$$ & $$6$$

Therefore $$k$$ MUST be divisible by $$3$$ & $$6$$

Option B
##### General Discussion
Senior SC Moderator
Joined: 22 May 2016
Posts: 3656
If k and m are integers and k = m*(m+4)*(m+5), k must be divisible by  [#permalink]

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19 Sep 2017, 08:47
2
1
1
petrified17 wrote:
If k and m are integers and k = m*(m+4)*(m+5), k must be divisible by which of following(s)?

I. 3
II. 6
III. 8

A. I
B. I and II
C. II and III
D. II
E. III

Plug in m = 1, m = 2, m = -7

(1)(5)(6) = 30
divisible by 3 and 6

(2)(6)(7) = 84
divisible by 3 and 6

(-7)(-3)(-2) = -42
divisible by 3 and 6

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Posts: 928
Re: If k and m are integers and k = m*(m+4)*(m+5), k must be divisible by  [#permalink]

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16 Oct 2018, 09:21
if m is divided by 3, so the expression is divided by 3
if m is not dividec by 3, m+1 or m+2 is divided by 3, and m+4 or m+5 is divided by 3.
in conclusion, the expression is divided by 3

m+4 and m+5 are consecutive numbers , so, one of them must be divided by 2
so, the expression is divided by 3

for 8
if m is divided by 2, but not divided by 4, m+4 is divided by 2. and m+7 is odd. so the expression is not divided by 8

Director
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Re: If k and m are integers and k = m*(m+4)*(m+5), k must be divisible by  [#permalink]

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17 Oct 2018, 00:20
if m is divided by 3, so the expression is divided by 3
if m is not dividec by 3, m+1 or m+2 is divided by 3, and m+4 or m+5 is divided by 3.
in conclusion, the expression is divided by 3

m+4 and m+5 are consecutive numbers , so, one of them must be divided by 2
so, the expression is divided by 3

for 8
if m is divided by 2, but not divided by 4, m+4 is divided by 2. and m+7 is odd. so the expression is not divided by 8

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Re: If k and m are integers and k = m*(m+4)*(m+5), k must be divisible by  [#permalink]

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17 Oct 2018, 19:03
petrified17 wrote:
If k and m are integers and k = m*(m+4)*(m+5), k must be divisible by which of following(s)?

I. 3
II. 6
III. 8

A. I
B. I and II
C. II and III
D. II
E. III

Roman Numeral I:

If m is 1, then k = 1 x 5 x 6.

If m is 2, k = 2 x 6 x 7.

If m is 3, k = 3 x 7 x 9.

In all three cases, we see that m*(m+4)*(m+5) must be divisible by 3. A more dynamic proof to show that k is divisible by 3 is to let m = 3q + r where q is an integer and r = 0, 1 or 2.

If m = 3q, obviously k is divisible by 3 since m is a multiple of 3.

If m = 3q + 1, then k is divisible by 3 since m + 5 = 3q + 6 is a multiple of 3.

If m = 3q + 2, then k is divisible by 3 since m + 4 = 3q + 6 is a multiple of 3.

Hence, k is always divisible by 3, regardless of the value of integer m.

Roman Numeral II:

We observe that m + 4 and m + 5 are two consecutive integers, hence, one of them is necessarily even. We know that k is even and from Roman Numeral I, we also know that k is divisible by 3. Thus, k must be divisible by 6.

Roman Numeral III:

We notice that m and m + 4 have the same parity (they are both odd or both even) and the parity of m + 5 is opposite to that of m (if m is even, m + 5 is odd and if m is odd, m + 5 is even). When we have a product of three integers, the product can be divisible by 8 if a) one of the integers is divisible by 8, b) one of the integers is divisible by 4 and one of the integers is divisible by 2 or c) each of the integers is divisible by 2.Since the parity of m + 5 is opposite to the remaining integers, it is impossible that each of m, m + 4 and m + 5 are divisible by 2. So if we pick a value for m such that neither of m, m + 4 and m + 5 are divisible by 4, the product cannot be divisible by 8. Such a value is m = 1, in which case k = 1*5*6 = 30. As we can see, k is not necessarily divisible by 8.

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Re: If k and m are integers and k = m*(m+4)*(m+5), k must be divisible by   [#permalink] 17 Oct 2018, 19:03
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