Progressive Intersections

\({AP}_{ 1 }=\quad 10,\quad 17,\quad 24,\quad 31,\quad \ldots\\
{ AP }_{ 2 }=\quad 9,\quad 14,\quad 19,\quad 24,\quad \ldots\\
{ AP }_{ 3 }= \quad 5,\quad 9,\quad 13,\quad 17,\quad\ldots\\
{ AP }_{ 4 }=\quad 5,\quad 8,\quad 11,\quad 14,\quad \ldots\)

Consider the four arithmetic progressions above. What is the smallest 4-digit number common to all these progressions?

This problem is a part of my set The Best of Me.

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