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# Data Sufficiency: Basic Work Order

If \$\$i\$\$ and \$\$d\$\$ are positive integers, what is the value of \$\$i\$\$? >(1) The remainder when \$\$i\$\$ is divided by \$\$(d+2)\$\$ is the same as when \$\$i\$\$ is divided by \$\$d\$\$. >(2) The quotient when \$\$i\$\$ is divided by \$\$(d+2)\$\$ is \$\$d\$\$.
Incorrect. [[snippet]] Even when combining both statements, the data in hand still isn't sufficient to determine a single definite value for \$\$i\$\$. Try proving this yourself: did your plug ins cover the entire range of possibilities for satisfying each statement?
According to Stat. (1+2), do the same numbers work for both statements? If \$\$d=2\$\$ and \$\$(d+2)=4\$\$, then - Stat. (1) allowed \$\$i\$\$ to be 5, 9, 13, as well as 4, 8, 12 and any other multiple of 4. - Stat. (2) allowed \$\$i\$\$ to be 8, 9, 10, 11. So both \$\$i=9\$\$ and \$\$i=8\$\$ are values that are on both lists. So there are two different i-values that satisfy both statements. So \$\$i\$\$ can be either 8 or 9 even when the statements are combined. No definite answer, so Stat.(1+2) → IS → E. Note that we managed to find two values of \$\$i\$\$ that satisfy both stats even before we started playing around with the value of \$\$d\$\$. For \$\$d=3\$\$, or 5, it is quite possible that we'd find even more values of \$\$i\$\$, further proving the statements insufficient.
For Stat. (2), try to use the same numbers. If \$\$d=2\$\$ and \$\$(d+2)=4\$\$, then Stat. (2) now reads: >(2) The quotient when \$\$i\$\$ is divided by 4 is 2. Which numbers give a quotient of 2 when divided by 4? Well, 4 times 2 is 8, so any number that is 8 or above (until 12) will give a quotient of 2 when divided by 4: 8, 9, 10, 11 all give a quotient of 2 when divided by 4 (they all have different remainders, but the quotient is still 2). So \$\$i\$\$ can be any 8, 9, 10, or 11, for \$\$d=2\$\$, and again there's no single value for \$\$i\$\$.
Incorrect. [[snippet]] According to Stat. (1), if \$\$d=2\$\$, then \$\$(d+2)=4\$\$. Stat. (1) now reads: >(1) The remainder when \$\$i\$\$ is divided by 4 is the same as when \$\$i\$\$ is divided by 2. Which numbers give the same remainder when divided by 4 as when divided by 2? The number 5 is an example: it gives a remainder of 1 when divided by 4 and by 2. But so will 9, or 13, or any number that is one more than a multiple of 4. There are even more numbers satisfying this constraint: 4, 8, and any other multiple of 4 leave a remainder of 0 when divided by both 2 and 4, so for \$\$d=2\$\$, \$\$i\$\$ could be any multiple of 4 as well. There's much more than one possible value for \$\$i\$\$ when \$\$d=2\$\$, so no definite answer for the value of \$\$i\$\$, and Stat.(1) → IS → BCE.
Incorrect. [[snippet]] According to Stat. (2), if \$\$d=2\$\$ and \$\$(d+2)=4\$\$, then Stat. (2) now reads: >(2) The quotient when \$\$i\$\$ is divided by 4 is 2. Which numbers give a quotient of 2 when divided by 4? Well, 4 times 2 is 8, so any number that is 8 or above (until 12) will give a quotient of 2 when divided by 4: 8, 9, 10, 11 all give a quotient of 2 when divided by 4 (they all have different remainders, but the quotient is still 2). So \$\$i\$\$ can be any 8, 9, 10, or 11, for \$\$d=2\$\$, and again there's no single value for \$\$i\$\$. No definite answer, so Stat.(2) → IS → CE.
Correct. [[snippet]] According to Stat. (1), if \$\$d=2\$\$, then \$\$(d+2)=4\$\$. Stat. (1) now reads: >(1) The remainder when \$\$i\$\$ is divided by 4 is the same as when \$\$i\$\$ is divided by 2. Which numbers give the same remainder when divided by 4 as when divided by 2? The number 5 is an example: it gives a remainder of 1 when divided by 4 and by 2. But so will 9, or 13, or any number that is one more than a multiple of 4. There are even more numbers satisfying this constraint: 4, 8, and any other multiple of 4 leave a remainder of 0 when divided by both 2 and 4, so for \$\$d=2\$\$, \$\$i\$\$ could be any multiple of 4 as well. There's much more than one possible value for \$\$i\$\$ when \$\$d=2\$\$, so no definite answer for the value of \$\$i\$\$, and Stat.(1) → IS → BCE.
For Stat. (2), try to use the same numbers. If \$\$d=2\$\$ and \$\$(d+2)=4\$\$, then Stat. (2) now reads: >(2) The quotient when \$\$i\$\$ is divided by 4 is 2. Which numbers give a quotient of 2 when divided by 4? Well, 4 times 2 is 8, so any number that is 8 or above (until 12) will give a quotient of 2 when divided by 4: 8, 9, 10, 11 all give a quotient of 2 when divided by 4 (they all have different remainders, but the quotient is still 2). So \$\$i\$\$ can be any 8, 9, 10, or 11, for \$\$d=2\$\$, and again there's no single value for \$\$i\$\$. No definite answer, so Stat.(2) → IS → CE.
Incorrect. [[Snippet]] According to Stat. (1), if \$\$d=2\$\$, then \$\$(d+2)=4\$\$. Stat. (1) now reads: >(1) The remainder when \$\$i\$\$ is divided by 4 is the same as when \$\$i\$\$ is divided by 2. Which numbers give the same remainder when divided by 4 as when divided by 2? The number 5 is an example: it gives a remainder of 1 when divided by 4 and by 2. But so will 9, or 13, or any number that is one more than a multiple of 4. There are even more numbers satisfying this constraint: 4, 8, and any other multiple of 4 leave a remainder of 0 when divided by both 2 and 4, so for \$\$d=2\$\$, \$\$i\$\$ could be any multiple of 4 as well. There's much more than one possible value for \$\$i\$\$ when \$\$d=2\$\$, so no definite answer for the value of \$\$i\$\$, and Stat.(1) → IS → BCE.
Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient to answer the question asked.
Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient to answer the question asked.
BOTH Statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.