Let S-10082-10072-10062-10052-10042- 10032- 62-52-42-32-22-12- then-Note- 1009 is a prime number-

S=1008^2 1005^2+1007^2 1004^2+1006^2 1003^2 +⋯ +6^2+5^2+4^2 3^2 2^2 1^2 =(2013)(3)+(2011)(3)+(2009)(3)+⋯ (3)(9)+(3)(7)+(3)(5) =(3)[5+7+9+11+⋯+2013] =3 ...

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Byju's Answer

Standard XII

Mathematics

Sum of n Terms

Let S=10082+1...

Question

Let $S = 1008^{2} + 1007^{2} + 1006^{2} - 1005^{2} - 1004^{2} - 1003^{2} + \dots 6^{2} + 5^{2} + 4^{2} - 3^{2} - 2^{2} - 1^{2}$ , then

[Note: $1009$ is a prime number]

S

is divisible by

67

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S

is divisible by

5

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The number of factors of

S

24

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The number of factors of

S

16

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Solution

The correct option is C The number of factors of $S$ is $24$
$S = 1008^{2} - 1005^{2} + 1007^{2} - 1004^{2} + 1006^{2} - 1003^{2} + \dots + 6^{2} + 5^{2} + 4^{2} - 3^{2} - 2^{2} - 1^{2}$
$= (2013) (3) + (2011) (3) + (2009) (3) + \dots (3) (9) + (3) (7) + (3) (5)$
$= (3) [5 + 7 + 9 + 11 + \dots + 2013] = 3 [1 + 3 + 5 + 7 + 9 + 11 + \dots + 2013 - 4] = 3 [1007^{2} - 4] = 3 (1009) (1005) = 3^{2} \times 5 \times 67 \times 1009$

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Similar questions

Q. Let

S = 1008^{2} + 1007^{2} + 1006^{2} - 1005^{2} - 1004^{2} - 1003^{2} + \dots 6^{2} + 5^{2} + 4^{2} - 3^{2} - 2^{2} - 1^{2}

, then

[Note:

1009

is a prime number]

Q. Let

S = 1008^{2} + 1007^{2} + 1006^{2} - 1005^{2} - 1004^{2} - 1003^{2} + \dots 6^{2} + 5^{2} + 4^{2} - 3^{2} - 2^{2} - 1^{2}

, then

[Note:

1009

is a prime number]

Q. Using the given pattern, find the missing numbers.

1^{2} + 2^{2} + 2^{2} = 3^{2}

2^{2} + 3^{2} + 6^{2} = 7^{2}

3^{2} + 4^{2} + 12^{2} = 13^{2}

4^{2} + 5^{2} + . . . . . . = 21^{2}

5^{2} + . . . . . + 30^{2} = 31^{2}

6^{2} + 7^{2} + . . . . . . = . . . . . . . .

Q. Using the given pattern, find the missing numbers:

1^{2} + 2^{2} + 2^{2} = 3^{2}

2^{2} + 3^{2} + 6^{2} = 7^{2}

3^{2} + 4^{2} + 12^{2} = 13^{2}

4^{2} + 5^{2} +

___

= 21^{2}

5^{2} +

___

+ 30^{2} = 31^{2}

6^{2} + 7^{2} +

___

=

____

Q. The value of

1^{2} - 2^{2} + 3^{2} - 4^{2} + 5^{2} - 6^{2} + . . . . + 99^{2} - 100^{2}

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Let S=10082+10072+10062−10052−10042− 10032+⋯62+52+42−32−22−12, then [Note: 1009 is a prime number]

The correct option is C The number of factors of S is 24S=10082−10052+10072−10042+10062−10032 +⋯+62+52+42−32−22−12 =(2013)(3)+(2011)(3)+(2009)(3)+⋯(3)(9)+(3)(7)+(3)(5) =(3)[5+7+9+11+⋯+2013]=3[1+3+5+7+9+11+⋯+2013−4]=3[10072−4]=3(1009)(1005)=32×5×67×1009

Let $S = 1008^{2} + 1007^{2} + 1006^{2} - 1005^{2} - 1004^{2} - 1003^{2} + \dots 6^{2} + 5^{2} + 4^{2} - 3^{2} - 2^{2} - 1^{2}$ , then

[Note: $1009$ is a prime number]