Edit: #4 below has been corrected. I am indebted to one of mathmom's astute students for catching my error!
Here are some suggestions for developing the foundation for today's challenge problem:
(1 ) List the first 5 positive integers which leave a remainder of 1 when divided by 2? Describe, in general, such positive integers.
(2) List the first 5 positive integers which leave a remainder of 3 when divided by 13? If you subtract 3 from each of these, what do you notice? Explain!
(3) List the first 5 positive integers which leave a remainder of 12 when divided by 13. If you subtract 12 from each of these, what do you notice? If, instead you ADD 1 to each of the 5 positive integers, what do you notice? Explain!
(4) What is the least positive integer N, greater than 1, which leaves a remainder of 1 when divided by 2, 3, 4 or 5? [Ans: 61]
Note: The word 'or' may be confusing or inaccurate here. Modify as needed!
Now for today's challenge (allow use of calculator):
What is the least positive integer which satisfies ALL of the following:
leaves a remainder of 1 when divided by 2
leaves a remainder of 2 when divided by 3
leaves a remainder of 3 when divided by 4
leaves a remainder of 4 when divided by 5
leaves a remainder of 5 when divided by 6
leaves a remainder of 6 when divided by 7
leaves a remainder of 7 when divided by 8
leaves a remainder of 8 when divided by 9.
This challenge looks harder than it is. Variations of these often appear on math contests for middle school and beyond. Simpler versions like example (4) above have appeared on the SATs.
Of course, modular arithmetic and congruences would make this problem trivial but that is non-standard and requires more time to develop.
I will not yet post the answer or possible solution...