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If f(n) = 1154*1156 for some integer, n, which of the following expres

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If f(n) = 1154*1156 for some integer, n, which of the following expres  [#permalink]

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New post 28 Nov 2019, 00:05
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A
B
C
D
E

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Question Stats:

35% (01:49) correct 65% (02:31) wrong based on 40 sessions

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Re: If f(n) = 1154*1156 for some integer, n, which of the following expres  [#permalink]

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New post 28 Nov 2019, 00:58
1
the last two digits = 24
options a,b,c,d wont be possible
IMO E ; 11n+10


If f(n)=1154∗1156f(n)=1154∗1156 for some integer, n, which of the following expressions could be equal to f(n)f(n)?


A. 3n−2

B. 5n−2

C. 5n+3

D. 7n−2

E. 11n+10
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Re: If f(n) = 1154*1156 for some integer, n, which of the following expres  [#permalink]

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New post 28 Nov 2019, 01:18
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1154 = 1155 - 1 = 11k - 1, for some positive value k
& 1156 = 1155 + 1 = 11k + 1

So, 1154*1156 = (11k - 1)(11k + 1) = (11k)^2 - 1 = 11(11k^2) - 1 = 11(n + 1) - 1 = 11n + 11 - 1 = 11n + 10
--> f(n) = 1154*1156 can be expressed of the form 11n + 10 for some value of 'n'

IMO Option E
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Re: If f(n) = 1154*1156 for some integer, n, which of the following expres  [#permalink]

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New post 28 Nov 2019, 02:01
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f(n)=1154∗1156 = 1155^2 -1

From options its visible that
1155 is divisible by 11....
so E: 1155^2 -1-10 = 1155^2 -11= which is divisible by 11

OA:E
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Re: If f(n) = 1154*1156 for some integer, n, which of the following expres  [#permalink]

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New post 28 Nov 2019, 05:11
If f(n)=1154∗1156 for some integer, n, which of the following expressions could be equal to f(n)?

For could be true questions, we only need to prove that the function is true for just one case.

1154*1156 = 1,324,024

Testing with f(n)=3n-2
3n-2=1,334,024
3n=1,334,022
n=444,674
Since f(n) is defined for integers and the function f(n)=3n-2 yields an integer when equated to 1154*1156, then f(n) is true for 3n-2, when n=444,674

The answer is option A.
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Re: If f(n) = 1154*1156 for some integer, n, which of the following expres  [#permalink]

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New post 28 Nov 2019, 06:22
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1154x1156= 1334024

Of the five answer choices, We notice that the last term is -2 in three of the options. But after adding -2, the resulting number 1334026 is not divisible by 5 or 3 or 7. Therefore eliminate A,B and D.

We add -10 to the number and notice that the result,1334014 is divisble by 11. (divisibility test : difference between sum of odd digits and even digits should be divisble by 11)

Therefore E is the answer
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Re: If f(n) = 1154*1156 for some integer, n, which of the following expres  [#permalink]

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New post 28 Nov 2019, 07:51
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Quote:
If f(n)=1154∗1156 for some integer, n, which of the following expressions could be equal to f(n)?

A. 3n−2
B. 5n−2
C. 5n+3
D. 7n−2
E. 11n+10


\(f(n)=1154*1156=1155^2-1^2=1155^2-1\)

A. \(3n−2=1155^2-1…3n=1155^2+1…1155=factor(3)…1≠factor(3)\)
B. \(5n−2=1155^2-1…5n=1155^2+1…1155=f(5)…1≠f(5)\)
C. \(5n+3=1155^2-1…1155=f(5)…-1-3=-4…-4≠f(5)\)
D. \(7n−2=1155^2-1…1155=f(7)…-1+2=1…1≠f(7)\)
E. \(11n+10=1155^2-1…1155=f(11)…-1-10=-11…-11=f(11)…f(n)=11n+10=valid\)

Ans (E)
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Re: If f(n) = 1154*1156 for some integer, n, which of the following expres  [#permalink]

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New post 28 Nov 2019, 08:45
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\(f(n)=1154∗1156=(1155-1)*(1155+1)=1155^2-1=(3^2*5^2*7^2*11^2)-1\)

We understand that the units digit of \(f(n)=1155^2-1\) must be 4.
(B) f(n)=5n−2 --> units digit of the result is either 3 or 8
(C) f(n)=5n+3 --> units digit of the result is either 3 or 8
Thus, we confidently eliminate choices (B) and (C), since both choices NEVER yield results with units digit of 4

A. f(n)=3n−2
\(f(n)=3n−2=(3^2*5^2*7^2*11^2)-1\)
\(3n=(3^2*5^2*7^2*11^2)+1\)
\(n=(3*5^2*7^2*11^2)+1/3\)
--> \(n\) is NOT an integer, so we confidently eliminate choice (A), since \(n\) has to be some integer

D. f(n)=7n−2
\(f(n)=7n−2=(3^2*5^2*7^2*11^2)-1\)
\(7n=(3^2*5^2*7^2*11^2)+1\)
\(n=(3^2*5^2*7*11^2)+1/7\)
--> \(n\) is NOT an integer, so we confidently eliminate choice (D), since \(n\) has to be some integer

E. 11n+10
\(f(n)=11n+10=(3^2*5^2*7^2*11^2)-1\)
\(11n=(3^2*5^2*7^2*11^2)-11\)
\(n=(3^2*5^2*7^2*11)-1\)
--> \(n\) is an integer, so we are confident that choice (E) is the CORRECT ANSWER


Final answer is (E)
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Re: If f(n) = 1154*1156 for some integer, n, which of the following expres  [#permalink]

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New post 28 Nov 2019, 13:02
IMO, Ans A.

Both 1154 and 1156 when divided by 3 leaves a remainder of 2.

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Re: If f(n) = 1154*1156 for some integer, n, which of the following expres  [#permalink]

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New post 28 Nov 2019, 17:09
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\(f(n) = 1154*1156= (1155 —1)(1155 +1)= 1155^{2}—1\)

\(1155^{2} —1= 11*11*105*105—1\)

E) \(11n + 10= 11n + 11—1=
11(n+1)—1\)

The answer is E

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Re: If f(n) = 1154*1156 for some integer, n, which of the following expres   [#permalink] 28 Nov 2019, 17:09
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