The sum of the first hundred positive integers is divisible by?

The sum of the first hundred positive integers is divisible by?

I. 2
II. 4
III. 8

The sum of the first hundred positive integers = 1 + 2 + ... + 100 = 100*101/2 = 50*101 = 5050 = 2*5*101*5

IMO A

Solution

• The sum of first hundred positive integer \(= \frac{100*(100+1)}{2} = 50*101 = 2*5^2*101\)
• Therefore, out of the given choices, the sum of first hundred positive integers is divisible by 2 only.

Thus, the correct answer is Option A.

n= total terms in sequence . a= first term l= last term
Sn = n/2 ( a+l)
= 100/2(1+100)
= 50(101)
= 5050

5050 is only divisible by 2
ANS:A

IMO A

S = (n/2)(first term+lastterm)

(100+1)×100/2

= 5050.

divisible by 2 onlyy

A

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Some relevant properties


i.e. Sum of 1 to 100, \(Sum = ∑100 = (\frac{1}{2})*100*(100+1) = 50*101\)

The sum of divisible by\( 2, 5^2\), and \(101\) but not by 4 and 8

Answer: Option A

The sum of the first hundred positive integers is divisible by?

I. 2
II. 4
III. 8

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
Solution:

Sum = \(\frac{(100)(2+99)}{(2)}\) = 5050

5050 only divisible by 2 out of given options

Answer choice A IMO

Solution:

The sum of the first hundred positive integers is:

(1 + 100)/2 x 100 = 101 x 50 = 5050

We see that 5050 is divisible by 2, but not by 4 and not by 8.

Answer: A

To determine if the sum of the first hundred positive integers is divisible by 2, 4, or 8, we can analyze the pattern of the sum.

The sum of the first n positive integers can be calculated using the formula: sum = n * (n + 1) / 2.

Let's consider each option:

I. 2: If a number is divisible by 2, it means it is even. The sum of the first hundred positive integers is 100 * (100 + 1) / 2 = 5050. Since 5050 is an even number, it is divisible by 2.

II. 4: If a number is divisible by 4, it means it is even and divisible by 2 twice. The sum of the first hundred positive integers is 5050. Dividing 5050 by 2 gives us 2525. Since 2525 is not divisible by 2, the sum is not divisible by 4.

III. 8: If a number is divisible by 8, it means it is even and divisible by 2 three times. The sum of the first hundred positive integers is 5050. Dividing 5050 by 2 repeatedly gives us 2525, 1262, and 631. Since 631 is not divisible by 2, the sum is not divisible by 8.

From the analysis, we can see that the sum of the first hundred positive integers is divisible by 2 (option I) but not divisible by 4 (option II) or 8 (option III).

Therefore, the answer is (A) I only.

how are we deriving the formula for Sum of first n positive integer is => n(n+1)/2?