Find the sum of largest five digit number with 3 distinct digit who is multiple of 11 and smallest four digit
number with 3 distinct digit who is multiple of 7.
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What is the largest 5-digit number which has distinct digits and is a multiple of 63?
What is the largest 5-digit number which has distinct digits and is a multiple of 63?Let N be a 5 -digit number, with distinct digits, which is divisible by both 7 and 9 . The largest N with distinct digits begins with 987 , and there are 7⋅6=42 5 -digit numbers with distinct digits starting with 987 . It is reasonable to expect that the largest N with distinct digits and a multiple of both 7 and 9 will come from this collection.
What is the largest 5-digit number which has distinct digits and is a multiple of 63?Let N be a 5 -digit number, with distinct digits, which is divisible by both 7 and 9 . The largest N with distinct digits begins with 987 , and there are 7⋅6=42 5 -digit numbers with distinct digits starting with 987 . It is reasonable to expect that the largest N with distinct digits and a multiple of both 7 and 9 will come from this collection.Let N=987ab be the desired 5 -digit number. Then a,b∈{0,1,2,…,6} and a≠b . Since 7∣987 , we must have 7∣ab . Also, 9∣N⇔9∣(a+b+24)⇔9∣(a+b−3) . This implies a+b=3 , for a+b≤6+5 . The only possibilities are 21 and 12 , and only the first is a multiple of 7 .
What is the largest 5-digit number which has distinct digits and is a multiple of 63?Let N be a 5 -digit number, with distinct digits, which is divisible by both 7 and 9 . The largest N with distinct digits begins with 987 , and there are 7⋅6=42 5 -digit numbers with distinct digits starting with 987 . It is reasonable to expect that the largest N with distinct digits and a multiple of both 7 and 9 will come from this collection.Let N=987ab be the desired 5 -digit number. Then a,b∈{0,1,2,…,6} and a≠b . Since 7∣987 , we must have 7∣ab . Also, 9∣N⇔9∣(a+b+24)⇔9∣(a+b−3) . This implies a+b=3 , for a+b≤6+5 . The only possibilities are 21 and 12 , and only the first is a multiple of 7 .The largest N satisfying the required conditions is 98721 . ■