Problem: The number which when divided by 10 leaves a remainder of 9, when divided by 9 leaves a remainder of 8, by 8 leaves a remainder of 7, etc., down to where, when divided by 2, it leaves a remainder of 1, is:
Answer Choices:
A. 59
B. 419
C. 1259
D. 2519
E. none of these answers
Solution:
Let N denote the number to be found; then we may write the given information as
N=10a9+9=9a8+8=⋯=2a1+1
Hence
N+1=10(a9+1)=9(a8+1)=⋯=2(a1+1)
ie., N+1 has factors 2,3,4,⋯,10 whose least common multiple is 23⋅32⋅5⋅7=2520. ∴N=2519.