Q) How many numbers lie between 10 and 300, which when divided by 4 leave a remainder 3? Also find their sum?

Ans: 

Step 1:

Since we need to find out numbers divisible by 4 between 10 and 300. These numbers will be in an AP where common difference will be 4.

Let’s find out 1st term and last term.

1st term will be the number which is higher than 10, divisible by 4 and eaves a remainder of 3 is 11

Last term of this AP is the number which is smaller than 300, divisible by 4 and eaves a remainder of 3 is 299

Step 2: Calculate number of terms in the AP:

We know that the nth term of an AP is given by, T_n  = a + (n – 1) d

Where, a = first term of the AP

d = common difference of the AP

Hence, n^th term, 299 = 11 + ( n – 1) 4

∴ 299 – 11 = (n – 1) 4

∴ 288 = (n – 1) 4

∴ (n -1) = = 72

∴ n = 73

∴ there will be 73 terms of this AP

Therefore, there will be 73 numbers between 50 and 500 which are divisible by 7.

Step 3: Calculating sum of these 73 terms:

We know that the sum of n terms is given by, S_n = [ 2 a + (n – 1) d ]

∴ S_n = [ 2 (11) + (73 – 1) (4) ]

∴ S₇₃ = (22 + 72 x 4) = x 310

∴ S₇₃ = 73 x 155 = 11,315

Therefore, sum of all integers is 11,315.

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