If p -1 - 1 -2 - 2 -3 - 3 -10 - 10 - then the remainder when p -2 is divided by 11 - is

P=1×1!+2×2!+3×3!+...+10×10! ⇒ p=(2 1)1!+(3 1)×2!+(4 1)×3!+...+(11 1)×10! ⇒ p=2! 1!+3! 2!+4! 3!+...+11! 10! ⇒ p=11! 1 ⇒ p+2=11!+1 When p+2 is divided by 11!, we ...

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Standard XII

Mathematics

Factorial

If p =1 × 1 !...

Question

If $p = 1 \times 1! + 2 \times 2! + 3 \times 3! + . . . + 10 \times 10!$ , then the remainder when $p + 2$ is divided by $11!$ is

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Solution

$p = 1 \times 1! + 2 \times 2! + 3 \times 3! + . . . + 10 \times 10!$
$\Rightarrow p = (2 - 1) 1! + (3 - 1) \times 2! + (4 - 1) \times 3! + . . . + (11 - 1) \times 10!$
$\Rightarrow p = 2! - 1! + 3! - 2! + 4! - 3! + . . . + 11! - 10!$
$\Rightarrow p = 11! - 1$
$\Rightarrow p + 2 = 11! + 1$
When $p + 2$ is divided by $11!$ , we get remainder as $1$

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If p=1×1!+2×2!+3×3!+...+10×10!, then the remainder when p+2 is divided by 11! is

p=1×1!+2×2!+3×3!+...+10×10! ⇒p=(2−1)1!+(3−1)×2!+(4−1)×3!+...+(11−1)×10! ⇒p=2!−1!+3!−2!+4!−3!+...+11!−10! ⇒p=11!−1 ⇒p+2=11!+1 When p+2 is divided by 11!, we get remainder as 1

If $p = 1 \times 1! + 2 \times 2! + 3 \times 3! + . . . + 10 \times 10!$ , then the remainder when $p + 2$ is divided by $11!$ is