Progressive Intersections

AP1=10,17,24,31,AP2=9,14,19,24,AP3=5,9,13,17,AP4=5,8,11,14,{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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