If m and n are non-zero integers, is m6-m5-m4 divisible by n?   1) m2

OFFICIAL SOLUTION

If you modify the original condition and the question,\(m^6-m^5-m^4 = nt\)? (t = any non-zero integer), so you get \(m^4(m^2-m-1)=nt\)?. If you look at con 1), from \(m^2-m-1=n^2\), if you substitute it to \(m^4(m^2-m-1)\), you get \(m^4(m^2-m-1)=m^4*n^2=n(m^4n)\), hence yes, it is sufficient. For con 2), it is difficult to approach the question, but if you apply CMT 4 (B: if you get A or B too easily, consider D), it is also sufficient. Therefore, the answer is D. If you actually solve it, for con 2), you get \((m^2)^2-2m(m^2)+m^2=(m^2-m)^2=n^2+11\), \((m^2-m)^2-n^2=(m^2-m-n)(m^2-m+n)=11\),

① When it becomes m^2-m-n=1 and m^2-m+n=11, from m^2-m-n=1, you get m^2-m-1=n, so m^4(m^2-m-1)=m^4*n, hence yes.

② When it becomes m2-m-n=11 and m^2-m+n=1, if you add the two equations, you get 2(m^2-m)=12, m^2-m=6 and n=-5, then m^2-m-1=6-1=5=(-1)(-5)=-n, so from m^4(m^2-m-1)=m^4(-n), it is also yes.

③ When it becomes m^2-m-n=-1 and m^2-m+n=-11, if you add the two equations, from 2(m^2-m)=-12, m^2-m=-6 and n=-5, integer m that satisfies this does not exist, so it is out of scope.

④ When it becomes m^2-m-n=-11 and m^2-m+n=-1, if you add the two equations, from 2(m^2-m)=-12, m^2-m=-6 and n=5, integer m that satisfies this does not exist, so it is out of scope.

Therefore, as ①, ② shown above, it always becomes yes, hence it is sufficient. Therefore, the answer is D.

Bunuel chetan2u VeritasKarishma Please help on solving St:2

Asked: If m and n are non-zero integers, is \(m^6-m^5-m^4\) divisible by n?

1) \(m^2-1=m+n^2\)
\(m^2 - m - 1 = n^2\)
\(m^4(m^2 - m - 1) = m^4*n^2\)
\(m^6-m^5-m^4 = m^4*n^2\)
\(m^6-m^5-m^4\) is divisible by n
SUFFICIENT

2) \(m^4-2m^3+m^2=n^2+11\)
\(m^4-2m^3+m^2 = m^2(m^2-2m+1) = m^2(m-1)^2 =n^2+11\)
\(m^6-m^5-m^4 = m^4(m^2-m-1) = m^4{(m-1)^2+m}\)
NOT SUFFICIENT

IMO A

Is \((m^4)(m^2-m-1)\) divisible by n?

Statement 2:
\(m^4-2m^3+m^2=n^2+11\)
\(m^2(m^2-2m+1)=n^2+11\)
\(m^2(m-1)^2=n^2+11\)
\(m^2(m-1)^2-n^2=11\)

Since m and n are integers, \(m^2(m-1)^2\) and \(n^2\) are two perfect squares with a difference of 11.
Make a list of perfect squares:
1, 4, 9, 16, 25, 36, 49...
Only 36 and 25 have a difference of 11.
Thus:
\(m^2(m-1)^2=36\)
\(n^2=25\) --> n=±5

\(m^2(m-1)^2=36\)
\(m(m-1)=±6\)

Case 1: \(m(m-1)=6\)
\(m^2-m=6\)
\(m^2-m-6=0\)
\((m-3)(m+2)=0\)
\(m=3\) or \(m=-2\)

Case 2: \(m(m-1)=-6\)
\(m^2-m=-6\)
\(m^2-m+6=0\)
No integer solutions.

If m=3, then \((m^4)(m^2-m-1)=81*5\)
If m=-2, then \((m^4)(m^2-m-1)=16*5\)
In each case, the resulting product is divisible by n=±5.
Thus, the answer to the question stem is YES.
SUFFICIENT.