GMAT 数学题（8）

小困蛇

这道题还没想出来，还想问一题有点擦边的
100！+1 的最大质因子是多少？ 谢谢

sdcar2010

是11

(a+b)^n-a^n-b^n必然有(a+b)的因子
根据 杨辉三角

-- by 会员 心中的永恒 (2011/1/4 15:36:47)

Only if n is an odd number.

Right answer.

sdcar2010

When n is an odd number,

(aⁿ + bⁿ) = (a+b)*(a^n-1 – a^n-2*b + a^n-3*b² – a^n-4*b³ + . . . – a²*b^n-3 + a*b^n-2 – b^n-1 )

So (a+b) is a factor of (aⁿ + bⁿ ) when n is odd.

sdcar2010

这道题还没想出来，还想问一题有点擦边的
100！+1 的最大质因子是多少？ 谢谢

-- by 会员 小困蛇 (2011/1/4 15:49:59)

No two consecutive positive integers(n, n+1) are ever divisible by same number except 1.
So all the factors of 100! is not a factor of 100!

100! = 100*99*98*97 . . . *50*49*48* . . . .*4*3*2*1

Its prime factors include all the prime numbers smaller than 100.

So the smallest prime factor for (100! + 1) is bigger than 100.  So it might be 101! As to the biggest prime factor of (100! + 1), I have no idea.