Answer :
The number on the selected card is [tex]4[/tex].
Given that:
[tex]1[/tex] collection = [tex]36[/tex] cards
[tex]36[/tex]cards = [tex]4[/tex] sets of [tex]9[/tex] cards per set
Each set contains cards that are numbered [tex]1-9[/tex]
Question => what is the number on the card that was removed?
Statement [tex]1[/tex] => The units digit of the sum of the numbers on the remaining [tex]35[/tex] cards is [tex]6[/tex]
Sum of digits on all [tex]9[/tex] cards = [tex]1 +2 +3 +4 +5 +6 +7 +8 +9 = 45[/tex]
Sum of all [tex]4[/tex] sets = [tex]4 * 45 = 180[/tex]
When one card is removed, the sum of the remaining [tex]35[/tex] cards has the last digit = [tex]6[/tex]
[tex]180 - 1 = 179, 180 - 2 = 178, 180 - 3 = 177, 180 - 4 = 176, 180 - 5 = 175, 180 - 6 = 174[/tex]
[tex]180 – 7 = 173, 180 – 8 = 172, 180 – 9 = 171[/tex]
[tex]180 - 4 = 176[/tex] is the only card that will have the sum of the remaining [tex]35[/tex] cards = [tex]176[/tex] (with the last digit being [tex]6[/tex])
Therefore, the number on the removed card = [tex]4[/tex] ; statement [tex]1[/tex] is SUFFICIENT
Statement [tex]2[/tex] => The sum of the numbers on the remaining [tex]35[/tex] cards is [tex]176[/tex]
Sum of digits on all [tex]9[/tex] cards = [tex]1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 +9 = 45[/tex]
Sum of all [tex]4[/tex] sets = [tex]4 * 45 = 180[/tex]
From this statement [tex]2[/tex] => [tex]180[/tex] - (value of selected card) = [tex]176[/tex]
[tex]180 - 176[/tex] = the value of the selected card
selected card = [tex]4[/tex]
Statement [tex]2[/tex] is SUFFICIENT
Because each statement by itself is ENOUGH
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