If p = 1! + (2x2!) + (3x3!) + (4x4!)...... + (10x10!), what is the
remainder when p+2 is divided by 11!?
Answer: 1
Explanation:
p = 1! + (2x2!) + (3x3!) + (4x4!)...... + (10x10!)
p = (1x1!) + (2x2!) + (3x3!) + (4x4!)...... + (10x10!)
But,
(1x1!) = 2! – 1!
(2x2!) = 3! – 2!
(3x3!) = 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.