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Question

Find the remainder when (17) (9!) + 2 (18!) is divided by (9!) 17408___

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Solution

[17(9!)+2(18!)]9!.17408=17.9!9!17.210+2.18!9!.17.210
17.9!9!17.210=1210and remainder of 1210=1
Hence, the Remainder of (17.9!9!17.210)=17.9!
2.18!=2.18.17.16.15.14.13.12.11.10.9!
In 18, 16, 14, 12, 10 the number of 2s are 1, 4, 1, 2 and 1 respectively.
So 2.18!=2.18.17.16.15.14.13.12.11.10.9!=210.17k.9!= Where k is an integer
Remainder (210.17k.9!9!.(17)210)=0
So the required remainder is 17.9!+0=17.9!

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