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本人已回答過這條問題
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when n=1, L.H.S=5^(2-1)-3^(2-1)-2^(2-1)=5^1-3^1-2^1=0
n=2,L.H.S=5^(4-1)-3^(4-1)-2^(4-1)=5^3-3^3-2^3=90, which is divisible by 15.
Assume n=k+1
5^(2k-1)-3^(2k-1)-2^(2k-1)=15N,where N is an integer.
when n=k+1
=[5^[2(k+1)-1)] - [3^[2(k+1)-1)] - [2^[2(k+1)-1)]
=[5^(2k+2-1)] - [3^(2k+2-1)] - [2^(2k+2-1) ]
=[5^(2k-1)(5^2)] - [3^(2k-1)(3^2)] - [2^(2k-1)(2^2)]
=[5^(2k-1)(25)] - [3^(2k-1)(9)] - [2^(2k-1)(4)]
=[5^(2k-1)(21+4)] - [3^(2k-1)(5+4)] - [2^(2k-1)(4)]
=[(21)(5^(2k-1))+(4)(5^(2k-1))] - [(5)(3^(2k-1))+(4)(3^(2k-1))] - [2^(2k-1)(4)]
remove bracket:
=(21)(5^(2k-1))+(4)(5^(2k-1)) - (5)(3^(2k-1)) - (4)(3^(2k-1)) - 2^(2k-1)(4)
arrange the common term in one side:
=(4)(5^(2k-1)) - (4)(3^(2k-1)) - (2^(2k-1))(4) + (21)(5^(2k-1)) - (5)(3^(2k-1))
take out the common term:
=4[5^(2k-1) - 3^(2k-1) - 2^(2k-1)] + (21)(5^(2k-1)) - (5)(3^(2k-1))
=4(15N) + (21)(5^(2k-1)) - (5)(3^(2k-1))
=4(15N) +(7)(3)(5)(5^(2k-2) - (3)(5)(3^(2k-2)
=4(15N) + (15)(7) (5^(2k-2) - (15)(3^(2k-2)
take out the common term:
=15[4N + (7) (5^(2k-2) - (3^(2k-2)]
By the principal of mathematical induction , P(n) is true for all positive integers n.

PROVE BY MATHEMATICAL INDUCTION
(5^2n-1)-(3^2n-1)-(2^2n-1)is divisible by 15
SOLUTION
Let P(n) be 5^(2n-1)-3^(2n-1)-2^(2n-1) is divisible by 15 for all positive integers n.

When n = 1,
5^(2*1-1)-3^(2*1-1)-2^(2*1-1)
=5-3-2
=0
which is divisible by 15
So P(1) is true

Assume P(k) is true, i.e. 5^(2k-1)-3^(2k-1)-2^(2k-1) is divisible by 15, i.e.
5^(2k-1)-3^(2k-1)-2^(2k-1) = 15m for a positive integer m

When n = k+1,
5^(2(k+1)-1)-3^(2(k+1)-1)-2^(2(k+1)-1)
=5^(2k+1)-3^(2k+1)-2^(2k+1)
=5^(2k-1+2)-3^(2k-1+2)-2^(2k-1+2)
=(5^2)(5^(2k-1)) - (3^2)(3^(2k-1)) - (2^2)(2^(2k-1))
=25(5^(2k-1)) - 9(3^(2k-1)) - 4(2^(2k-1))
=4(5^(2k-1) - 3^(2k-1) - 2^(2k-1)) + 21(5^(2k-1)) - 5(3^(2k-1))
=4(15m) + 21(5^(2k-1)) - 5(3^(2k-1))
=15(4m) + 7*3*5*(5^(2k-2)) - 5*3*(3^(2k-1))
=15(4m) + 15(7*5^(2k-2)) - 15(3^(2k-1))
=15(4m + 7*5^(2k-2) - 3^(2k-1))
which means it's divisible by 15
So P(k+1) is true.

By Mathematical induction, 5^(2n-1)-3^(2n-1)-2^(2n-1) is divisible by 15 for all positive integers n.


小小補充：
這條題目最難的地方應該是上面綠色的部份，要能抽出 15 的因子。

因為 4 不是 15 (3 和 5) 的因子，所以只好保留所有的 4 項，從 3 和 5 的因子項著手。