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if a(1)b(1)c(1), a(2)b(2)c(2)" and " a(...

if `a_(1)b_(1)c_(1), a_(2)b_(2)c_(2)" and " a_(3)b_(3)c_(3)` are three-digit even natural numbers and `Delta = |{:(c_(1),,a_(1),,b_(1)),(c_(2),,a_(2),,b_(2)),(c_(3),,a_(3),,b_(3)):}|" then " Delta ` is

A

divisible by 2 but not necessarily by 4

B

divisible by 4 but not necessarily by 8

C

divisible by 8

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
As ` a_(1) ,b_(1),C_(1) , a,b,c " and " a_(3) ,b_(3) ,c_(3) ` are even natural numbers , each of `C_(1),C_(2),C_(3)` is divisible by 2, Let `C_(i) = 2k_(i) " for " i=1,2,3` thus
`Delta = 2 |{:(k_(1),,a_(1),,b_(1)),(k_(2),,a_(2),,b_(2)),(k_(3),,a_(3),,b_(3)):}|=2m`
Where m is some natural number. thus `Delta ` is divisible by 2. That `Delta ` may not be divisible by 4 can be seen by taking the three numbers as 112, 122 and 134 Note that
`Delta =|{:(2,,1,,1),(2,,1,,2),(4,,1,,3):}|=2`
Which is divisible by 2 but not 4
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