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Numbers - Unit digit

SOURCE:COMPETITION
Number of Problems: 2.
FOR PRINT ::: (Book)

Problem Num : 1

Type:

Topic:Numbers

Adjustment# : 0

Difficulty: 1

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What is the tens digit in the sum $7!+8!+9!+...+2006!$
$mathrm{(A) } 1qquad mathrm{(B) } 3qquad mathrm{(C) } 4qquad mathrm{(D) } 6qquad mathrm{(E) } 9$
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Category Unit digit

Analysis

Solution/Hint
Since $10!$ is divisible by $100$ , any factorial greater than $10!$ is also divisible by $100$ . The last two digits of all factorials greater than $10!$ are $00$ , so the last two digits of $10!+11!+...+2006!$ is $00$ . (*)
So all that is needed is the tens digit of the sum $7!+8!+9!$
$7!+8!+9!=5040+40320+362880=408240$
So the tens digit is $4 Rightarrow C$
(*) A slightly faster method would have to take the $pmod {100}$ residue of $7! + 8! + 9!.$ Since $7! = 5040,$ we can rewrite the sum as $5040 + 8cdot 5040 + 72cdot 5040 equiv 40 + 8cdot 40 + 72cdot 40 = 40 + 320 + 2880 equiv 40 pmod{100}.$ Since the last two digits of the sum is $40$ , the tens digit is $4.$

Problem Num : 2

Type:

Topic:Numbers

Adjustment# : 0

Difficulty: 1

'
What is the hundreds digit of $2011^{2011}?$
$extbf{(A) } 1 qquad extbf{(B) } 4 qquad extbf{(C) }5 qquad extbf{(D) } 6 qquad extbf{(E) } 9$
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Category Unit digit

Analysis

Solution/Hint