Problem Solving (PS) | How many positive integers less than 500 have a remainder of 1 when

Let's assume that the number is\(n\)

...a remainder of 1 when divided by7...

\( n = 7q_1 +1\)

\(q_1\) → quotient when n is divided by\(7\)

...a remainder of 2 when divided by3...

\( n = 3q_2 +2\)

\(q_2\) → quotient when n is divided by\(3\)

Both the equations can be merged into a single equations

\( n = \text{LCM}(7,3)q + \text{first commonterm}\)

\(q\) → quotient when n is divided by\(21\)

To find the first common term, let's write a few of the terms of each of the sequences-

\( n = 7q_1 +1\) ⇒\( 1, 8, 15, 22, 29, 36, ....\)

\( n = 3q_2 +2\) ⇒\( 2, 5, 8, 11, ....\)

Hence, the first common term =8

\( n = 21q +8\)

This equation represents an arithmetic progression.

The first term of the equation is\(8\) , i.e. when\( q =0\)

Last term =

\( 500 = 21q +8\)

\( 21q =492\)

\( q =23.XX\)

Number of terms =\( 23 - 0 + 1 =24\)

OptionC