Thread: [SOLVED] Difficult binomial theorem question(involving greatest integer function)?

If (3√3 + 5)^9 = R and [R] denotes the greatest integer less than or equal to R, then:

A) [R] is divisible by 10
B) [R^2] is divisible by 512
C) [R] is divisible by 15
D) [R] is an even number

More than one options may be correct.

Hello again fardeen_gen

As with your previous posting, I'm not sure how you're supposed to tackle this question. I have used a calculator, and found [R] to be 1191041440. So A and D are true. The others are not.

Grandad

Originally Posted by fardeen_gen

If (3√3 + 5)^9 = R and [R] denotes the greatest integer less than or equal to R, then:

A) [R] is divisible by 10
B) [R^2] is divisible by 512
C) [R] is divisible by 15
D) [R] is an even number

More than one options may be correct.

Here is another approach:




Subtracting, the even-numbered terms cancel, yielding



So



because
and the summation is an integer.

Originally Posted by awkward

Here is another approach:




Subtracting, the even-numbered terms cancel, yielding



So



because
and the summation is an integer.

That is pretty neat! It's clear, then, that R is a multiple of 10. And it's not a multiple of 3, since all the terms are except the last. Therefore R is not a multiple of 15. Any idea how this might determine the truth of statement B? (Other than by evaluating [R], of course.)

Grandad

PS I have just looked again at statement B, and noticed that it is , not just R. Since , this will be true if R is a multiple of 32. And it is. So B is true after all!