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==> If you modify the original condition and the question, in order to get xy=even, it can either be x=even? or y=even?. For con 1), from (x,y)=(odd,even),(even,odd), you always get yes, hence it is sufficient. For con 2), you always get x=even, hence it is yes and sufficient.

John travelled for 12hrs. What is the total average speed?

1) For the first 8hrs, he travelled 60miles per 1hr. 2) For the last 8hrs, he travelled 50miles per 1hr.

==> In the original condition, he is travelling 12 hours, and for con 1) and con 2), it is divided into 8hrs each. Thus, the distance of first 4hrs, the second 4hrs, and the last 4hrs are unknown, hence there are too many variables.

When n is divided by 4, the remainder is 3. If 3^n+2 is divided by 5, what is the remainder?

A. 1 B. 2 C. 3 D. 4 E. 0

==> You get n=4p+3, which becomes 3^n+2=3^{4p+3}+2=(3^4)^p(3^3)+2=(~1)^p(27)+2=(~1)(27)+2=(~7)+2=~9. Thus, the remainder when it is divided by 5 becomes 4.

==> If you modify the original condition and the question, from is p/m>0?, you multiply m^2 on both sides, you get (Squared number is always positive, so even if you multiply, the inequality sign doesn't change) p/m(m^2)>0(m^2)?, which becomes pm>0?.

If a, b, and c are positive integers, is (a+b)c divisible by 3?

1) 2-digit integer ab is divisible by 3. 2) When c is divided by 3, the remainder is 0.

==> If you modify the original condition and the question, in order to have (a+b)c to be divided by 3, a+b or c has to be divided by 3. However, if you look at con 2), c is divisible by 3, hence it is yes and sufficient. For con 1), ab is also divisible by 3, and thus a+b is also divisible by 3, hence it is yes and sufficient. Therefore, the answer is D. This type of question is a 5051-level question which applies CMT 4 (B: if you get A or B too easily, consider D).

There are 5 apples in a bag. 4 apples are good but 1 apple is rotten. If you take out 2 apples from the bag, what is the probability that 1 apple selected is rotten?

A. 1/3 B. 1/4 C.2/5 D. 3/5 E.1/7

==> From probability=want/total, want=selecting one rotten apple and one good apple, and total=selecting 2 of 4 apples, so you get probability=4C1*1C1/5C2=(4)(1)/(5*4/2!)=4/10=2/5.

When A works alone, it takes 14hrs, and when A works with B together, it takes 10hrs. How many hours does it take B to work alone?

A. 30hrs B. 33hrs C. 35hrs D. 37hrs E. 40hrs

==> For work rate questions, you solve “together and alone” reciprocally. It takes A 14hrs alone, and if you set the hours it took B to work alone as B hrs, from (1/14)+(1/B)=1/10 and 1/B=(1/10)-(1/14)=1/35, you get B=35.

==> If you modify the original condition and the question, in order to get |x/y|=x/y, you get x/y≥0?, and if you multiply y^2 on both sides, you get xy≥0?. Then, for con 1), to satisfy |xy|=xy, you get xy≥0, hence yes, it is sufficient. For con 2), if x=y=2, yes, but if x=2 and y=-1, no, hence it is not sufficient.

==> In the original condition, there are 2 variables (x,y) and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), for con 2), you get xy=1, hence it is unique and sufficient. For con 1), from (xy)^2-xy=0, xy(xy-1)=0, you get xy=0,1, but since x and y are positive, you get xy=1, and thus con 1) = con 2).

What is the remainder, when n(n+2) is divided by 24 for a positive integer x?

1) n is an even integer 2) n has remainder 0 or 1 when it is divided by 3.

=>Condition 1) There are two kinds of even integers, which are n = 4k or n = 4k + 2 for some integer k. That means n could have 0 or 2 as a remainder when it is divided by n. If n = 4k, n(n+2) = 4k(4k+2) = 8k(2k+1) and n(n+1) is a multiple of 8. If n= 4k+2, n(n+2) = (4k+2)(4k+2+2) = 2(2k+1)*4(k+1) = 8(2k+1)(k+1) and n(n+1) is a multiple of 8. For both cases, n(n+1) is a multiple of 8. However, we can’t identify the remainder when it is divided by 3 from the condition 1).

Condition 2) n = 3k or n = 3k +1. If n = 3k, n(n+2) = 3k(3k+2) is a multiple of 3. If n = 3k + 1, n(n+2) = (3k+1)(3k+3) = 3(3k+1)(k+1) is a multiple of 3. Thus, for both cases, n(n+1) is a multiple of 3. However, we can’t identify the remainder when it is divided by 8 from the condition 2).

Condition 1) & 2) From the condition 1), n(n+1) is a multiple of 8. And n(n+1) is a multiple of 3 from the condition 2). Therefore, n(n+1) is a multiple of 24 from the both conditions 1) & 2) together.

==> In the original condition, there are 2 variables (x,y) and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), for con 1), from x^4+y^4-2x^2y^2=0, (x^2-y^2)2=0, you get x^2=y^2, and from x=±y, yes and no coexists, hence it is not sufficient. For con 2), you only get x=y=0, hence yes, it is sufficient.

Is the total profit from the sales of 3 products greater than $6?

1) The least profit from the sales of the 3 products is at least $2.1 2) The 2nd largest profit from the sales of the 3 products is at least $3.1.

=>Condition 1) Since the minimum profit of 3 products is $2.1, the total profit of them is greater than or equal to $2.1 x 3 = $6.3, which is greater than $6. Thus this is sufficient.

Condition 2) Since the 3nd largest profit is at least $3.1, the sum of the first and the second products is at least $6.2, which is greater than $6. Thus this is sufficient too.

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