If Line K in the XY-Plane has equation y=mx+b, where m and b are constants, what is the slope of K?
1. K is parallel to the line with equation y=(1-m)x+(b+1)
2. K intersects the line with equation y=2x+3 at the point (2,7)
Answer: A
From statement (1): y=(1-m)x+(b+1) has the same slope as y=mx+b. (Parallel lines have same slope)
Thus 1-m = m
implies Slope of K=m=1/2 ---- Hence sufficient
From statement (2): just says line y=2x+3 is not parallel to K, these two lines can have any angle between them other than 0, 180, 360 degrees ---- hence insufficient
Hence answer A
Sunday, April 27, 2008
Data Sufficiency - 46
What is the greatest common divisor of positive integers a and b?
(1) a and b share exactly one common factor
(2) a and b are both prime numbers
Answer: A
From statement (1): we know that a and b have only one common factor, and we also know that all positive integers share the common factor 1 only, so we know it must be 1...hence sufficient
From Statement (2): we know that a and b are both prime, this implies the greatest common factor will have to be 1 or if a = b could be the same prime number then the GCF would be a (=b). ...hence insufficient
NOTE: You cannot assume that a and b are different integers if the question stem does not states the same
(1) a and b share exactly one common factor
(2) a and b are both prime numbers
Answer: A
From statement (1): we know that a and b have only one common factor, and we also know that all positive integers share the common factor 1 only, so we know it must be 1...hence sufficient
From Statement (2): we know that a and b are both prime, this implies the greatest common factor will have to be 1 or if a = b could be the same prime number then the GCF would be a (=b). ...hence insufficient
NOTE: You cannot assume that a and b are different integers if the question stem does not states the same
Sunday, April 20, 2008
Problem Solving - 47
If n is a positive integer and the product of all the integers from 1 to n, inclusive, is a multiple of 990, what is the least possible value of n?
A) 10
B) 11
C) 12
D) 13
E) 14
Answer: B
990 is a multiple of n! implies it must contain all the prime factors of 990
Largest prime factor of 990 is 11 implies n! must have 11 as a factor
Now since n! = 990x where x is integer it implies it can have prime factors more than 11 but not less than 11
Thus least possible value of n is thus 11
A) 10
B) 11
C) 12
D) 13
E) 14
Answer: B
990 is a multiple of n! implies it must contain all the prime factors of 990
Largest prime factor of 990 is 11 implies n! must have 11 as a factor
Now since n! = 990x where x is integer it implies it can have prime factors more than 11 but not less than 11
Thus least possible value of n is thus 11
Thursday, April 10, 2008
Problem Solving - 46
A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for its dessert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different dessert platters could the restaurant offer?
A) 8
B) 12
C) 15
D) 21
E) 27
Answer: E
Kinds of platter:
1 cheese + 1 fruit
Total = 6 * 2 = 12 types of platters
2 cheese + 2 fruit
Total = 6C2 * 2C2 = 15 * 1 = 15 types
Total: 12 + 15 = 27
A) 8
B) 12
C) 15
D) 21
E) 27
Answer: E
Kinds of platter:
1 cheese + 1 fruit
Total = 6 * 2 = 12 types of platters
2 cheese + 2 fruit
Total = 6C2 * 2C2 = 15 * 1 = 15 types
Total: 12 + 15 = 27
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