Data Sufficiency: Yes/No Basic Technique

If $$a$$, $$b$$, and $$i$$ are integers, is $$2b + 1 = a$$? >(1) If $$i$$ is divided by 4, then the quotient is $$a$$ and the remainder is 3. >(2) If $$i$$ is divided by 8, then the quotient is $$b$$ and the remainder is 7.
According to Stat. (1+2), >$$i = 4a + 3$$ >$$i = 8b + 7$$ Since both expressions equal $$i$$, you can set them equal to one another: >$$4a + 3 = 8b + 7$$ Then see if you can change this equation into $$2b+1 = a$$ algebraically: >$$4a = 8b + 4$$ >$$a = 2b + 1$$ This is a definite answer, so **Stat.(1+2) → Yes → S → C**. __Alternative explanation__: Test the problem out by __Plugging In__ numbers. The problem is finding a value of $$i$$ that satisfies both statements. To overcome that, prepare a list of numbers for each case and see which numbers are on both lists: * Remainder of 3 when divided by 4: 7, 11, 15, 19, 23, 27, and 31 * Remainder of 7 when divided by 8: 15, 23, and 31 So the numbers that satisfy both statements are 15, 23, and 31. Let's test these out. For each of them, what is $$a$$? What is $$b$$? Is $$2b+1=a$$? Variables $$a$$ and $$b$$ are the quotients when dividing by 4 and 8, respectively. If you plug in $$i=15$$, then you get * When 15 is divided by 4, the quotient is $$\frac{12}{4}=3$$, so $$a = 3$$. * When 15 is divided by 8, the quotient is $$\frac{8}{8}=1$$, so $$b = 1$$. In this case, you get >$$2b + 1 = 2(1)+1=3$$, so $$2b + 1 = a$$ is true, giving an answer of "__Yes__." Try it out for each of the examples above (i.e. $$i = 23$$, and $$i = 31$$). You should find that $$2b+1$$ always equals $$a$$. After trying these three examples, you can see that the combined statements are sufficient with a high degree of certainty.
Incorrect. [[snippet]] According to Stat. (1), >$$i = 4a + 3$$. Even if you find a definite value for $$a$$ from this equation, $$b$$ can have any value. There is no definite answer, so **Stat.(1) → No → IS → BCE**. According to Stat. (2), >$$i = 8b + 7$$. Even if you find a definite value for $$b$$ from this equation, $$a$$ can have any value. There is no definite answer, so **Stat.(2) → No → IS → CE**.
Incorrect. [[snippet]] According to Stat. (1+2), >$$i = 4a + 3$$ >$$i = 8b + 7$$ Since both expressions equal $$i$$, you can set them equal to one another: >$$4a + 3 = 8b + 7$$ Then see if you can change this equation into $$2b+1 = a$$ algebraically: >$$4a = 8b + 4$$ >$$a = 2b + 1$$ This is a definite answer, so **Stat.(1+2) → Yes → S → C**.
Incorrect. [[snippet]] According to Stat. (2), >$$i = 8b + 7$$. Even if you find a definite value for $$b$$ from this equation, $$a$$ can have any value. There is no definite answer, so **Stat.(2) → No → IS**.
Incorrect. [[snippet]] According to Stat. (1), >$$i = 4a + 3$$. Even if you find a definite value for $$a$$ from this equation, $$b$$ can have any value. There is no definite answer, so **Stat.(1) → No → IS → BCE**.
Correct. [[snippet]] According to Stat. (1), >$$i = 4a + 3$$. Even if you find a definite value for $$a$$ from this equation, $$b$$ can have any value. There is no definite answer, so **Stat.(1) → No → IS → BCE**. According to Stat. (2), >$$i = 8b + 7$$. Even if you find a definite value for $$b$$ from this equation, $$a$$ can have any value. There is no definite answer, so **Stat.(2) → No → IS → CE**.
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.
EACH statement ALONE is sufficient to answer the question asked.
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
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