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Math and Logic

Getting Lines Crossed

If this pattern continued, how many more lines of numbers would we need to add to see two lines intersect on a circled number?

Keep reading to find out, or jump ahead to the challenge for a tougher question.

To answer this question, we need to unravel any patterns in the figure. Breaking the figure apart a few pieces at a time might help, so let's start with the solid lines and the circled numbers they pass through.

Every circled number is a multiple of eight, and there aren't any uncircled multiples of eight in the figure. If we added another row of numbers, the next circled number should be 7×8=56.7 \times 8 = 56. Extending one of the solid lines takes us right through 56:56:

Now, let's focus on the dashed lines and the circled numbers they go through. Before we added another row of numbers, the dashed lines only passed through three circled numbers. Adding the 5656 gives us a fourth circled number we can extend a dashed line through — this is because the dashed lines go through all multiples of 14:14:

The first circled number both kinds of lines go through is 56,56, since it is the least common multiple of 88 and 14:14:

What if we went all the way to 500?500? How many times would lines intersect on circled numbers?

Today's Challenge

If this pattern continued up to the number 500,500, how many times would lines intersect on circled numbers?

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