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Show that 1!+2!+3!++n ! cannot be a perf...

Show that `1!+2!+3!++n !` cannot be a perfect square for any `n in N ,ngeq4.`

Text Solution

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We have,
`1!+2!+3!+..+n!=1+2+6+24+5!+6!+..+n!`
`=33+5!+6!+..+n!`
`=33+10k, k in 1`
This sum always ends with 3 or the digit in the unit's place of the sum is 3. Now, we know that square of any integer never ends with 3. Then the given sum cannot be a perfect square.
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