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05 Apr 2022, 08:30
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D
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Difficulty:
95%
(hard)
Question Stats:
50% (02:38) correct
50%
(02:21)
wrong
based on 147
sessions
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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
(A) 21
(B) 22
(C) 23
(D) 24
(E) 25
14 Apr 2022, 01:08
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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
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
General Discussion
06 Nov 2022, 04:25
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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
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
