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

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Competition Mode Question

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. $$11n + 10$$

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

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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

Originally posted by chondro48 on 28 Nov 2019, 07:45.
Last edited by chondro48 on 13 Jan 2020, 07:01, edited 2 times in total.
##### General Discussion
GMAT Club Legend  V
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Re: If f(n) = 1154*1156 for some integer, n, which of the following expres  [#permalink]

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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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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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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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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

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

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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)

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

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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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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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$$f(n) = 1154*1156= (1155 —1)(1155 +1)= 1155^{2}—1$$

$$1155^{2} —1= 11*11*105*105—1$$

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

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

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f(n)=1154∗1156=(1155−1)∗(1155+1)=1155^2−1

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

For all options, put ->
3n - 2 = 1155^2−1
or
5n - 2 = 1155^2−1
and so on.

Except for E - 11n+10, you'll get a fraction for every other answer and since n is an integer, E is the answer.

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

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1154/3, Remainder= 2
1156/3, Remainder= 1
So, 1154*1156/3, Remainder= 2*1= 2
Therefore 3n+2 or 3n-1 can be the functions and not 3n-2 as we need to remove this Remainder to make it divisible by 3.

Similarly,
1154/5, Remainder= 4
1156/5, Remainder= 1
So, 1154*1156/5, Remainder= 4*1= 4
Therefore 5n+4 or 5n-1 can be the functions and not 5n-2 or 5n+3 as we need to remove this Remainder to make it divisible by 5.

Similarly,
1154/7, Remainder= 6
1156/7, Remainder= 1
So, 1154*1156/7, Remainder= 6*1= 6
Therefore 7n+6 or 7n-1 can be the functions and not 7n-2 as we need to remove this Remainder to make it divisible by 7.

Now,
1154/11, Remainder= 10
1156/11, Remainder= 1
So, 1154*1156/11, Remainder= 10*1= 10
Therefore 11n+10 or 11n-1 can be the function as we need to remove this Remainder to make it divisible by 11.

Hence, 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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chondro48 You deserve way better than Q49 chondro48 wrote:
$$f(n)=1154∗1156=(1155-1)*(1155+1)=1155^2-1=(3^2*5^2*7^2*11^2)-1$$

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

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

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I do agree.
Some of his explanation are very good.
nick1816 wrote:
chondro48 You deserve way better than Q49 chondro48 wrote:
$$f(n)=1154∗1156=(1155-1)*(1155+1)=1155^2-1=(3^2*5^2*7^2*11^2)-1$$

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

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