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If the remainder when positive integer x is divided by 7 is

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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Thu Dec 20, 2018 5:47 am
AAPL wrote:If the remainder when positive integer x is divided by 7 is 4, what is the value of x?
1) x is less than 50
2) x is prime
Target question: What is the value of x?

Given: If the remainder when positive integer x is divided by 7 is 4
When it comes to remainders, we have a nice rule that says:
If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.
From the given information, some possible values of x are: 4, 11, 18, 25, 32, 39, 46, 53, 60, 67, . . . etc

Statement 1: x is less than 50
The possible values of x are: 4, 11, 18, 25, 32, 39, 46, 53, 60, 67, . . . etc.
So, statement 1 tells us that x could equal 4, 11, 18, 25, 32, 39 or 46
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x is prime
The possible values of x are: 4, 11, 18, 25, 32, 39, 46, 53, 60, 67, . . . etc.
From the list, we can already see two prime numbers.
x could equal 11 or 53
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
11 is the only possible value of x that satisfies BOTH statements.
So, the answer to the target question is x = 11
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent
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by fskilnik@GMATH » Thu Dec 20, 2018 1:13 pm
AAPL wrote:Veritas Prep

If the remainder when positive integer x is divided by 7 is 4, what is the value of x?

1) x is less than 50
2) x is prime
$$x \ge 1\,\,{\mathop{\rm int}} \,\,\,\left( * \right)$$
$$x = 7M + 4\,\,\,,\,\,M\mathop \ge \limits^{\left( * \right)} 0\,\,\,{\mathop{\rm int}} $$
$$? = x$$

$$\left( 1 \right)\,\,x < 50\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,x = 4\,\,\,\,\, \hfill \cr
\,{\rm{Take}}\,\,x = 4 + 7 = 11 \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{INSUFF}}.$$
$$\left( 2 \right)\,\,x\,\,{\rm{prime}}\,\,\,\left\{ \matrix{
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,x = 11\,\,\,\,\, \hfill \cr
\,{\rm{Take}}\,\,x = 11 + 6 \cdot 7 = 53 \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{INSUFF}}{\rm{.}}$$
$$\left( {1 + 2} \right)\,\,\,\,? = 11\,\,\,\,\,\,\left( {{\rm{11}}\,{\rm{and}}\,\,{\rm{53}}\,\,{\rm{are}}\,\,{\rm{the}}\,\,{\rm{smallest}}\,\,{\rm{possibilities}}\,\,{\rm{for}}\,\,{\rm{the}}\;{\rm{bifurcation}}\,\,{\rm{of}}\,\,\left( {\rm{2}} \right)} \right)$$


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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