Pairing high

Number Theory Level 1

Let $x$ and $y$ be positive integers less than 10. There are 81 ordered pairs $(x, y)$ , from $(1, 1)$ to $(9, 9)$ . Since they are ordered pairs, $(1, 3)$ is considered different from $(3, 1)$ .

How many of these 81 ordered pairs have the property that $x \times y$ is at least 50?

The answer is 10.

1 solution

Denton Young
Jan 4, 2016

The largest number has to be at least 8, because the largest product we can get by only using numbers up to 7 is 7 * 7 = 49, which is less than 50.

If the largest number is 9: 9 * 5 = 45, so the other number has to be at least 6. That yields 7 ordered pairs: (9, 6), (6, 9), (9, 7), (7, 9), (9, 8), (8, 9) and (9, 9).

If the largest number is 8: 8 * 6 = 48, so the other number has to be at least 7. that yields 3 more ordered pairs: (8, 7), (7, 8) and (8, 8).

7 + 3 = 10, so there are 10 ordered pairs with the required property.

Moderator note:

Great! Conditioning on the larger number makes it easier to approach this count.

Pairing high

The answer is 10.

1 solution

Moderator note:

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