How many numbers less than 500 exist such that when any of these is di

Question

How many positive integers less than 500 exist such that when any of these is divided by 7 the remainder is 3 and when any of these is divided by 3 the remainder is 1?

(A) 21
(B) 22
(C) 23
(D) 24
(E) 25

rajatchopra1994 · Accepted Answer

Explanation:

Explanation:
x = 7a + 3
x = 3b + 1

First digit which gives remainder 3 when divided by 7 and remainder 1 when divided by 3 is 10.

x = 21c + 10

x must be less than 500, so
21c + 10 < 500
21c < 490
c < 23.3

As, c can be any integer between 0 and 23, so total = 24 values

IMO-D

BrushMyQuant · Answer

We need to find How many positive integers less than 500 exist such that when any of these is divided by 7 the remainder is 3 and when any of these is divided by 3 the remainder is 1?

When the number (lets say n) is divided by 7 the remainder is 3

Theory: Dividend = Divisor*Quotient + Remainder

n -> Dividend
7 -> Divisor
a -> Quotient (Assume)
3 -> Remainder

=> n = 7*a + 3 = 7a + 3 ...(1)

When the number is divided by 3 the remainder is 1

Let quotient = b

=> n = 3b + 1 ...(2)
=> 7a + 3 = 3b + 1
=> 3b = 7a + 2

=> b =  \frac{7a + 2}{3} 
Only those values of "a" which will give "b" also as an integer will give us the common values for the integer "n".

=> a = 1, 4, 7,... (so values of "a" are starting form 1 and increasing by 3)
(This follows a pattern of 3k - 2 ,  Watch this video  of  Sequence to learn how to find this ))

Now n has to be ≤ 500
=> 7a + 3 ≤ 500
=> 7a ≤ 497
=> a ≤ 71
=> 3k - 2 ≤ 71
=> 3k ≤ 73
=> k ≤ 24.3

=> 24 value of a or 24 values of n are possible

So,  Answer will be D 
Hope it helps!

Watch the following video to learn the Basics of Remainders

https://www.youtube.com/watch?v=P0S6j2uTaj4

adewale223 · Answer

You can also create a list...

Let the number be X

When X divided by 7 remainder is 3.
X = 3, 10, 17, 24...

When X divided by 3 remainder is 1

X = 1, 4, 7, 10, 13...

So first number common in list is 10.

LCM of 3 and 7 is 21 so numbers are

10, 31, 52, 73, 94....493.

493-10 = 483/21 
23+1 = 24.

Posted from my mobile device

Wadree · Answer

LCM of 3 and 7 is 21......... and only number less than 21 that leaves remainder 1 & 3 when divided by 3 & 7 respectively is 10....... in other words among every 21 numbers the 1Oth number matches required condition.... example 31, 52...... so 500/21=23.something and 23x21=483 so 17 is left which means 1 more number matches condition and dat is 493......... so total number dat match condition 23+1=24

How many numbers less than 500 exist such that when any of these is di

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How many numbers less than 500 exist such that when any of these is di

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