Let S -10082-10072-10062-10052-10042- 10032- 62-52-42-32-22-12- then-Note- 1009 is a prime number-A- S is divisible by 67B- S is divisible by 5C- The number of factors of S is 24D- The number of factors of S is 16

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Sum of n Terms

Let S =10082+...

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

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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}

1009

α

x^{2} - x + 1 = 0.

S = {(α + \frac{1}{α})}^{2} + {(α^{2} + \frac{1}{α^{2}})}^{2} + {(α^{3} + \frac{1}{α^{3}})}^{2} + \dots + {(α^{100} + \frac{1}{α^{100}})}^{2} .

If a number 'x' is divisible by another number 'y', then 'x' is also divisible by all the prime factors of 'y'.

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F

(i)

(i i)

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(i v)

2

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