Number Properties – Factorial & Remainders

If p = 1! + (2×2!) + (3×3!) + (4×4!)…… + (10×10!), what is the remainder when p+2 is divided by 11!?

Answer: 1

Explanation:

p = 1! + (2×2!) + (3×3!) + (4×4!)…… + (10×10!)

p = (1×1!) + (2×2!) + (3×3!) + (4×4!)…… + (10×10!)

But,

(1×1!) = 2! – 1!

(2×2!) = 3! – 2!

(3×3!) = 4! – 3! And so on.

Hence,

p = 2! – 1! + 3! – 2! + 4! – 3! + 5! – 4! + 6! – 5! + 7! – 6! + 8! – 7! + 9! – 8! + 10! – 9! + 11! – 10!

= 11! – 1!

Hence p + 2 = 11! – 1! + 2

= 11! + 1

Now 11! + 1 when divided by 11! will leave a remainder 1.

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