Problem of the Week 8

Two measuring sticks use the same unit, but the scale on one stick has 635 marks per unit, and the scale on the other stick has 465 marks per unit.  Suppose the sticks are placed together so that the 0 marks on each scale line up.  What is the number of the next mark on the 465 scale that lines up exactly with a mark on the 635 scale?

 

Alternate Problem 8

If the pattern in the following 5 by 5 square is completed,what is the sum of all the numbers?

 

2

 

3
5
8
13

 

4

 

7
11

 

6

 

10

 

10

 

27

 

16

 

 

Extension Problem 8

Consider two different arithmetic sequences: 3, 14, 25, 36, 47, 58, 69, 80, …, where the numbers increase successively by 11, and 2, 9, 16, 23, 30, …, where the numbers increase successively by 7.  What are the first 10 numbers which appear in both sequences?