Question

11. a) Write the first 3 digit number which leaves a remainder 2 on division by 5
b) Which is the last such three digit number?
c) Write the sequence of three digit numbers which leaves a remainder 2 on division by 5
d) How many three digit numbers are there which leaves a remainder 2 on division by 5​

Answer 1

Answer:

The answer of a and b part are as follows:-

a) 102

b) 997

Answer 2

$\large\underline{\sf{Solution-}}$

Let first find the series of three digit natural numbers which leaves a remainder 2 when divided by 5.

So, such three digit natural numbers are

$\rm :\longmapsto\:102, \: 107, \: 112, \: - - - - , \: 997$

So, it means

Answer (a) First 3 digit natural number which leaves remainder 2 when divided by 5 is 102

Answer (b) Last 3 digit natural number which leaves remainder 2 when divided by 5 is 997.

Answer (c) Sequence of 3 digit natural numbers which leaves remainder 2 when divided by 5 is

$\rm :\longmapsto\:102, \: 107, \: 112, \: - - - - , \: 997$

Answer (d)

Since, the above sequence forms an Arithmetic Progression with

• First term, a = 102

• Common difference, d = 107 - 102 = 5

• nth term = 997

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

Tʜᴜs,

$\rm :\longmapsto\:997 = 102 + (n - 1)5$

$\rm :\longmapsto\:997 - 102 = (n - 1)5$

$\rm :\longmapsto\:895 = (n - 1)5$

$\rm :\longmapsto\:n - 1 = 179$

$\rm \implies\:\boxed{ \tt{ \: n \: = \: 180 \: }}$

So, it means there are 180 three digit natural numbers which leaves remainder 2 when divided by 5.

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More to Know :-

↝ Sum of n  terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• Sₙ is the sum of n terms of AP.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.