Question

In a certain deck of cards, each card has a positive integer written on it. In a multiplication game, a child draws a card and multiplies the integer on the card by the next larger integer. If each possible product is between 15 and 200, then the least and greatest integers on the cards could be

Answer 1

4 and 13

Explanation:

You want integer solutions to ...

15 ≤ n(n+1) ≤ 200

If we let the limits be represented by "a", then the equality is represented by ...

n² +n -a = 0

(n² +n +1/4) -a -1/4 = 0

(n +1/2)^2 = (a +1/4)

n = -1/2 + √(a +1/4)

For a=15, we have

n ≥ -1/2 + √15.25 ≈ 3.4 . . . . . minimum n is 4

For a=200, we have

n ≤ -1/2 + √200.25 ≈ 13.7 . . . maximum n is 13

The least and greatest integers on the cards are 4 and 13.