If K is a positive integer less than 10 and N=4, 321+K, what is the value of K?

(1)N is divisible by 3.

(2)N is divisible by 7.

If n is an integer, is n/15 an integer?
/>(1)3n/15is an integer.
(2)8n/15 is an integer.

Given a positive number N, when N is rounded by a certain method (for convenience, call it Method Y), the result is 10ⁿ if and only if n is an integer and 5×10^n-1≤N<5×10ⁿ. In a certain gas sample, there are, when rounded by Method Y, 10²¹molecules of H₂ and also 10²¹ molecules of O₂. When rounded by Method Y, what is the combined number of H₂ and O₂ molecules in the gas sample?

(1)The number of H₂ molecules and the number of O₂ molecules are each less than 3 X10²¹.

(2)The number of H₂ molecules is more than twice the number of O₂ molecules.

If m and n are positive integers, is n even?

(1)m(m+2)+1=mn

(2)m(m+n)is odd.

n=2⁴·3²·5²and positive integer d is a divisor of n.Is d>~$\sqrt{n}$~ ?

(1)d is divisible by 10.

(2)d is divisible by 36.

The first four digits of the six-digit initial password for a shopper's card at a certain grocery store is the customer's birthday in day-month digit form. For example, 15 August corresponds  to 1508 and 5 March corresponds to 0503. The 5th digit of the initial password is the units digit of seven times the sum of the first and third digits, and the 6th digit of the initial password is the units digit of three times the sum of the second and fourth digits. What month, and what day of that month, was a customer born whose initial password ends in 16?

(1)The customer's initial password begins with 21, and its fourth digit is 1.

(2)The sum of the first and third digits of the customer's initial password is 3, and its second digit is 1.

If @ denotes a mathematical operation, does x@y=y@x for all x and y?

(1)For all x and y, x@y=2(x²+y²).

(2)For all y,0@y=2y².

The 9 participants in a race were divided into 3 teams with 3 runners on each team. A team was awarded 6-n points if one of its runners finished in nth place, where 1≤n≤5. If all of the runners finished the race and if there were no ties was each team awarded at least 1 point?

(1)No team was awarded more than a total of 6 points.

(2)No pair of teammates finished in consecutive places among the top five places.

If x and y are integers, is xy+1 divisible by 3?

(1)When x is divided by 3, the remainder is 1.

(2)When y is divided by 9, the remainder is 8.

If x is a positive integer, how many positive integers less than x are divisors of x?

(1)x² is divisible by exactly 4 positive integers less than x².

(2)2x is divisible by exactly 3 positive integers less than 2x.