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# If a is a positive integer and if remainders of 4 and 6 are obtained

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Math Expert
Joined: 02 Sep 2009
Posts: 49913
If a is a positive integer and if remainders of 4 and 6 are obtained  [#permalink]

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22 Feb 2018, 20:44
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15% (low)

Question Stats:

82% (01:53) correct 18% (01:57) wrong based on 84 sessions

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If a is a positive integer and if remainders of 4 and 6 are obtained when 89 and 125, respectively, are divided by a, then a =

(A) 7

(B) 9

(C) 15

(D) 17

(E) 19

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If a is a positive integer and if remainders of 4 and 6 are obtained  [#permalink]

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22 Feb 2018, 22:13
Bunuel wrote:
If a is a positive integer and if remainders of 4 and 6 are obtained when 89 and 125, respectively, are divided by a, then a =

(A) 7

(B) 9

(C) 15

(D) 17

(E) 19

From the above question stem, we get the following equations
xa + 4 = 89 -> (1)
ya + 6 = 125 -> (2)

(1) -> xa = 85
(2) -> ya = 119

Dividing (2) by (1) we get $$\frac{x}{y} = \frac{5}{7}$$

If x=5 and y=7, then the value of a is 17(Option D)
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Re: If a is a positive integer and if remainders of 4 and 6 are obtained  [#permalink]

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23 Feb 2018, 01:15
Bunuel wrote:
If a is a positive integer and if remainders of 4 and 6 are obtained when 89 and 125, respectively, are divided by a, then a =

(A) 7

(B) 9

(C) 15

(D) 17

(E) 19

We'll go one option at a time and eliminate impossible answers.
This is an Alternative approach.

If 89/a gives a remainder of 4 that means that a divides 89 - 4 = 85.
(A) divides 84, (B) divides 90, (C) divides 90 and (E) divides 76 (and 95) so none of them divide 85.

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If a is a positive integer and if remainders of 4 and 6 are obtained  [#permalink]

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23 Feb 2018, 02:07
Bunuel wrote:
If a is a positive integer and if remainders of 4 and 6 are obtained when 89 and 125, respectively, are divided by a, then a =

(A) 7

(B) 9

(C) 15

(D) 17

(E) 19

check with options

89 mod 7 != 4
89 mod 9 != 4
89 mod 15 != 4
89 mod 17 = 4
125 mod 17 = 6

(D) imo
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Re: If a is a positive integer and if remainders of 4 and 6 are obtained  [#permalink]

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23 Feb 2018, 05:07
1
Bunuel wrote:
If a is a positive integer and if remainders of 4 and 6 are obtained when 89 and 125, respectively, are divided by a, then a =

(A) 7

(B) 9

(C) 15

(D) 17

(E) 19

When 89 is divided by a then the remainder is 4
i.e. 89-4=85 must be divisible by a

When 125 is divided by a then the remainder is 6
i.e. 125-6=119 must be divisible by a

i.e. a must be a factor of 85 as well as of 119

HCF of 85 and 119 = 17

Hence a = 17

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Re: If a is a positive integer and if remainders of 4 and 6 are obtained  [#permalink]

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15 Mar 2018, 06:47
Bunuel wrote:
If a is a positive integer and if remainders of 4 and 6 are obtained when 89 and 125, respectively, are divided by a, then a =

(A) 7

(B) 9

(C) 15

(D) 17

(E) 19

89 = nq + 4

$$\frac{85}{q}$$= n -----I

125 = np + 6

$$\frac{119}{p}$$ = n -----II

From I & II, n =$$\frac{85}{q}$$= $$\frac{119}{p}$$

17 is the number which divides both 85 & 119.

(D)
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Re: If a is a positive integer and if remainders of 4 and 6 are obtained  [#permalink]

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15 Mar 2018, 07:00
Best solution would be the substitution of answer options.

Posted from my mobile device
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Re: If a is a positive integer and if remainders of 4 and 6 are obtained  [#permalink]

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15 Mar 2018, 08:13
Bunuel wrote:
If a is a positive integer and if remainders of 4 and 6 are obtained when 89 and 125, respectively, are divided by a, then a =

(A) 7

(B) 9

(C) 15

(D) 17

(E) 19

Max possible Divisors are 85 & 119 , leaving remainder of 4 & 6

Find the HCF of 85 & 119 : 1 & 17

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If a is a positive integer and if remainders of 4 and 6 are obtained  [#permalink]

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15 Mar 2018, 09:57
Bunuel wrote:
If a is a positive integer and if remainders of 4 and 6 are obtained when 89 and 125, respectively, are divided by a, then a =

(A) 7

(B) 9

(C) 15

(D) 17

(E) 19

let quotients=q and p
a=85/q=119/p
17 is only common factor
D
If a is a positive integer and if remainders of 4 and 6 are obtained &nbs [#permalink] 15 Mar 2018, 09:57
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