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190

Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$, of play blocks which satisfies the conditions:

  • If $16$, $15$, or $14$ students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and
  • There are three integers $0 < x < y < z < 14$ such that when $x$, $y$, or $z$ students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.

Find the sum of the distinct prime divisors of the least possible value of $N$ satisfying the above conditions.


225
Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 12n = 2012$.

773
A class collects $50$ dollars to buy flowers for a classmate who is in the hospital. Roses cost $3$ dollars each, and carnations cost $2$ dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly $50$ dollars?

905
A line passes through $A\ (1,1)$ and $B\ (100,1000)$. How many other points with integer coordinates are on the line and strictly between $A$ and $B$?

948

Soda is sold in packs of $6$, $12$ and $24$ cans. What is the minimum number of packs needed to buy exactly $90$ cans of soda?


Joe wants to measure $6$ liter water using just two containers whose capacities are $27$ liters and $15$ liters, respectively. Can you help him?

A class collects $50$ dollars to buy flowers for a classmate who is in the hospital. Roses cost $3$ dollars each, and carnations cost $2$ dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly $50$ dollars?

Of the pairs of positive integers $(x, y)$ that satisfies $3x+7y=188$, which ordered pair has the least positive difference $x-y$?

Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 16n = 2016$