Open Middle®
Directions: Using the digits 0 to 9 at most one time each, fill in the boxes to create a three-digit number. Try to create a three-digit number divisible by the greatest (or fewest) amount of the following factors: 2, 3, 4, 5, 6, 8, 9 or 10.
Hint
What number should the final digit be such that it is divisible by 2, 5 and 10?
What should the three digits add up to so that it is divisible by 3 and 9?
What should the middle digit be so that it is divisible by 4?
Answer
Greatest number of factors: 360 and 720 are both divisible by 2,3,4,5,6,8,9 and 10.
Fewest number of factors: any three digit prime with unique digits would have none of the given numbers as factors. There are more than 20 such numbers (e.g. 103, 347, 521, 849).
Fewest number of factors: any three digit prime with unique digits would have none of the given numbers as factors. There are more than 20 such numbers (e.g. 103, 347, 521, 849).
Source: Kelly Zinck
I believe the answer drop down box is wrong as 360 and 740 are divisible by 2, 3, 4, 5, 6, 8, 9, and 10. The six is missing form the drop down menu answer. Thanks for the cool problem.
Thank you for that catch Brad! The correction has been made.
740 is NOT divisible by 3…or 6…or 8…or 9!!
Problem states 720 not 740.
you should put 180 in “2, 3, 4, 5, 6, 8, 9”
720
810
360- can be divided by 2, 3,4,6, 9and 10
360 has the greatest
The number 360 is divisible by them all.
720
divisible by all
360 is the least number divisible by all.
720
720
Always a good day to learn something new! 😉
360 and 720
There are 240 numbers not divisible by 2, 3 or 5 (and therefore not divisible by 4, 6, 8, 9 or 10 either) and not all of them are prime numbers (for example, 133 = 7 * 19):
101
103
107
109
113
119
121
127
131
133
137
139
143
149
151
157
161
163
167
169
173
179
181
187
191
193
197
199
203
209
211
217
221
223
227
229
233
239
241
247
251
253
257
259
263
269
271
277
281
283
287
289
293
299
301
307
311
313
317
319
323
329
331
337
341
343
347
349
353
359
361
367
371
373
377
379
383
389
391
397
401
403
407
409
413
419
421
427
431
433
437
439
443
449
451
457
461
463
467
469
473
479
481
487
491
493
497
499
503
509
511
517
521
523
527
529
533
539
541
547
551
553
557
559
563
569
571
577
581
583
587
589
593
599
601
607
611
613
617
619
623
629
631
637
641
643
647
649
653
659
661
667
671
673
677
679
683
689
691
697
701
703
707
709
713
719
721
727
731
733
737
739
743
749
751
757
761
763
767
769
773
779
781
787
791
793
797
799
803
809
811
817
821
823
827
829
833
839
841
847
851
853
857
859
863
869
871
877
881
883
887
889
893
899
901
907
911
913
917
919
923
929
931
937
941
943
947
949
953
959
961
967
971
973
977
979
983
989
991
997
By the way, the solution given lists 849, but 849 is divisible by 3.
I wonder why 7 is left out and 10 has been included.