If x is an integer, which of the following must be an odd integer?
If xy ≠ 0, is x/y = 1?
(1) x^{2} = y^{2}
(2) xy > 0
The value of (9×10^{7})(9×10^{8}) is closest to which of the following?
The product of the units digit, the tens digit, and the hundreds digit of the positive integer m is 96. What is the units digit of m?
1) m is odd. ?
2) The hundreds digit of m is 8. ?
If m, p, and t are positive integers and m < p < t, is the product mpt an even integer?
1) t – p = p  m
2) t – m = 16
If x and y are positive integers, and x^{3}y^{4} = 2,000, which of the following is the value of xy?
If ab = 1, what is the value of (axb)(ayb)?
1) ax = by = 2
2) 2xy = 4
If positive integer x is a multiple of 6 and positive integer y is a multiple of 14, is xy a multiple of 105?
(1) x is a multiple of 9. ?
(2) y is a multiple of 25. ?
The sum of all the integers k such that –26 < k < 24 is
If a, b, c, and d are positive integers, is (a/b) (c/d) > c/b?
1) c > b
2) a > d
If k, m, and p are integers, is k – m – p odd?
1) k and m are even and p is odd.?
2) k, m, and p are consecutive integers.
Is x^{2} + y^{2} > 6?
1) (x + y)^{2} > 6
2) xy = 2
The numbers x and y are threedigit positive integers, and x + y is a fourdigit integer. The tens digit of x equals 7 and the tens digit of y equals 5. If x < y, which of the following must be true?
I. The units digit of x + y is greater than the units digit of either x or y.
II. The tens digit of x + y equals 2.
III. The hundreds digit of y is at least 5.
If n is an integer and 100 < n < 200, what is the value of n?
1) n/36 is an odd integer.
2) n/45 is an even integer.
If x and y are positive integers and 1 + x + y + xy = 15, what is the value of x + y?
If x and y are positive integers and 1 + x + y +xy = 21, what is the value of x?
1) y > 3
2) y = 6
What is the sum of a certain pair of consecutive odd integers?
1) At least one of the integers is negative.
2) At least one of the integers is positive.
If x and y are integers, is x + y greater than 0?
1) x is greater than 0.
2) y is less than 1.
If x and y are consecutive even numbers, what is the sum of x and y?
1) x is negative.
2) y is nonpositive.
If a and b are positive integers such that a – b and a/b are both even integers, which of the following must be an odd integer?
Is atleast one of x and z odd?
1) x + z = odd
2) x – z = odd
If x and z are integers, is atleast one of x and z odd?
1) x + z = odd
2) x – z = odd
What is the greatest common factor of the positive integers j and k?
(1) k = j + 1?
(2) jk is divisible by 5.
If x and y are positive integers and y = √(9 −?x), what is the value of y?
1) x < 8
2) y > 1
If x and y are integers and y = √(9 −?x), what is the value of y?
1) x < 11
2) y > 0
In the decimal representation of x, where 0 < x < 1, is the tenths digit of x nonzero?
1) 16x is an integer. ?
2) 8x is an integer. ?
If x and z are integers, is at least one of them even?
1) x + z is even.
2) x – z is even.
What is the remainder when the twodigit, positive integer x is divided by 3?
1) The sum of the digits of x is 5.?
2) The remainder when x is divided by 9 is 5.
Statement 1: The sum of the digits of x is 5.
Divisibility rule of 3:
If the sum of the digits of a number is divisible by 3, then the number is divisible by 3.
However, if the sum of the digits of a number is NOT divisible by 3, then the remainder is the same when the number is divided by 3.
Thus, the remainder when the twodigit, positive integer x is divided by 3 = Remainder when 5 is divided by 3 = 2. Sufficient.
Statement 2: The remainder when x is divided by 9 is 5.
The rule for divisibility of 9 is same as that for 3.
The remainder when the twodigit, positive integer x is divided by 3 = Remainder when 5 is divided by 3 = 2. Sufficient.
Alternatively,
Say X = 9q + 5, where q is quotient
Thus, X/3 = (9/3)*q + 5/3
X/3 = 3q + 1 + 2/3
=> Remainder = 2. Sufficient.
What is the remainder when the twodigit, positive integer x is divided by 9?
1) The sum of the digits of x is 5.?
2) The remainder when x is divided by 3 is 2.
A school administrator will assign each student in a group of n students to one of m classrooms. If 3 < m < 13 < n, is it possible to assign each of the n students to one of the m classrooms so that each classroom has the same number of students assigned to it?
1) It is possible to assign each of 3n students to one of m classrooms so that each classroom has the same number of students assigned to it. ?
2) It is possible to assign each of 13n students to one of m classrooms so that each classroom has the same number of students assigned to it. ?
What is the value of x?
(1) 5 < x^{2 }< 10?
(2) x is an integer.
For any positive integer n, the sum of the first n positive integers equals [n(n+1)]/2. What is the sum of all the even integers be tween 99 and 301?
A positive integer n is said to be “primesaturated” if the product of all the different positive prime factors of n is less than the square root of n. What is the greatest twodigit primesaturated integer?
If the units digit of the threedigit positive integer k is nonzero, what is the tens digit of k?
1) The tens digit of k + 9 is 3.
2) The tens digit of k + 4 is 2.
If k is negative, which of the following must also be negative?
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