Question

Find the remainder when 111+222+333+444+...+999 is divided by 55? A) 1 B) none of these C) 45 D) 54

Answer 1

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✦ Required Answer:

❇ GiveN:

• Number = 111+222+333+444....+999
• This is to be divided by 55.

❇ To FinD:

• What would be the remainder when the number is divided by 55....?

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✦ How to solve?

The question might be looking tough, but it's quite simple. Just observation is needed, here 111, 222, 333...999 all are divisible by 111, So taking 111 common, we can simplify to get the sum without actual division. Then, simply divide the no. with 55 to get the remainder.

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✦ Solution:

The given number/sum,

111 + 222+ 333 + 444 +.....+ 999 can be written as

111 + 2 ×111+ 3×111 +4×111.....+ 9×111

Taking 111 as common,

111(1+2+3....9)

So, now we can use sum of n consecutive terms formula, that is derived from Snth formula, that is

Sum of n terms starting from 1 upto n

$\large{ \boxed{ \rm{ = \frac{n(n + 1)}{2} }}}$

By using formula,

$\large{ \rm{ \longrightarrow \: 111( \dfrac{9 \times 10}{2} }}) \\ \\ \large{ \rm{ \longrightarrow \: 111 \times 45}}$

So, finally our simplified number is 111 × 45.

Now, it is divided by 55.

By using Euclid's division lemma,

Dividend = Divisor × Quotient + Remainder

$\large{ \rm{ \longrightarrow \: 111 \times 45 = 55 \times q + r}} \\ \\ \large{ \rm{ \longrightarrow \: 4995 = 55 \times q + r}}$

After division, we will get

$\large{ \rm{ \longrightarrow \: 4995 = 55 \times 90 + 45}}$

Hence, our quotient is 90

and Remainder is 45.

Answer -$\large{\boxed{\red{\rm{ 45}}}}$

✏Option - C

$\large{ \therefore{ \underline{ \underline{ \rm{ \pink{Hence, \: solved \: \dag}}}}}}$

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Answer 2

Question :

Find the remainder when 111+222+333+444+...+999 is divided by 55.

Solution:

We have to find remainder when 111+222+333+444+..999 is divide by 55

First Solve :

111+222+333+444+.....999

Cleary this in Airthemtic progression:

Here, a = 111

$\sf\:a_{n}=999$

n = no of terms = 9

$\rm{S_{n}=\dfrac{n}{2}\times[a+a_{n}]}$

$\implies\rm{S_{n}=\dfrac{9}{2}\times[111+999]}$

$\implies\rm{S_{n}=\dfrac{9}{2}\times[1110]}$

$\implies\rm{S_{n}=4995}$

Now , when 111+222..+....+999 is divided by 55

⇒4995 divided by 55.

We know that ,

Dividend = Divisor ×Quotient + Remainder

49995= 55×90+45

Therefore,the remainder is 45