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Question 30

The product of four positive integers $$a,b,c,d$$ is $$9!$$. The numbers $$a,b,c,d$$ satisfy $$ab+a+b=1224$$, $$bc+b+c=549$$ and $$cd+c+d=351$$. Then $$a+b+c+d$$ is


Correct Answer: 108

Solution

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Rewrite the three conditions as $$(a+1)(b+1)=1225=49\times25$$, $$(b+1)(c+1)=550=25\times22$$, and $$(c+1)(d+1)=352=22\times16$$. Hence $$a+1=49$$, $$b+1=25$$, $$c+1=22$$ and $$d+1=16$$, giving $$a=48,b=24,c=21,d=15$$. Their sum is $$48+24+21+15=108$$.

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