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QOTD#14 When x is divided by 3, the remainder is 1. When x

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sandy
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QOTD#14 When x is divided by 3, the remainder is 1. When x [#permalink] New post   14 Nov 2016, 15:14
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When x is divided by 3, the remainder is 1. When x is divided by 7, the remainder is 2. How many positive integers less than 100 could be values for x?


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Re: QOTD#14 When x is divided by 3, the remainder is 1. When x [#permalink] New post   14 Nov 2016, 15:21
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Explanation

To solve this question, write it out.

Since there are fewer numbers that yield a remainder of 2 when divided by 7, start there. The first such number is 2, and thereafter they increase by 7; the rest of the list is thus 9, 16, 23, 30, 37, 44, 51, 58, 65, 72, 79, 86, and 93. Rather than list out all the numbers that yield a remainder of 1 when divided by 3, just select the numbers that meet the requirement from the list you already have: Only 16, 37, 58, and 79 do, so there are 4 values for x.

Hence the correct answer is 4.
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Re: QOTD#14 When x is divided by 3, the remainder is 1. When x [#permalink] New post   25 Nov 2016, 17:46
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sandy wrote:
When x is divided by 3, the remainder is 1. When x is divided by 7, the remainder is 2. How many positive integers less than 100 could be values for x?



When it comes to remainders, we have a nice rule that says:

If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

Now onto the question....
When x is divided by 7, the remainder is 2
Possible values of x are: 2, 9, 16, 23, 30, 37, 44, 51, 58, 65, 72, 79, 86, 93

When x is divided by 3, the remainder is 1
Possible values of x are: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97

The two sets have 4 numbers in common.
Answer:
Show: ::
4


ASIDE: Notice that each of the shared values (16, 37, 58, and 79) are 21 greater than the previous shared value. Also notice that 21 is the least common multiple (LCM) of 3 and 7.
So, once we found 1 value in common, we could have just kept adding 21 to that value to find the subsequent values.

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Re: QOTD#14 When x is divided by 3, the remainder is 1. When x [#permalink] New post   22 Jan 2018, 08:19
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in 1st case when N=4/3..remainder=1
2nd case when N=9/7...remainder=2
looking at condition we can conclude that 3,9,27,81.....in short, the power of 3 is increasing there and the limit is till 100 hence last number to consider is 81...making it total 4 numbers.
answer 4
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Re: QOTD#14 When x is divided by 3, the remainder is 1. When x [#permalink] New post   20 Oct 2022, 08:36
Given that When x is divided by 3, the remainder is 1. When x is divided by 7, the remainder is 2. And we need to find How many positive integers less than 100 could be values for x

Theory: Dividend = Divisor*Quotient + Remainder

When x is divided by 3, the remainder is 1

x -> Dividend
3 -> Divisor
a -> Quotient (Assume)
1 -> Remainders
=> x = 3*a + 1 = 3a + 1

When x is divided by 7, the remainder is 2

x -> Dividend
7 -> Divisor
b -> Quotient (Assume)
2 -> Remainders
=> x = 7*b + 2 = 7b + 2

x = 3a + 1 = 7b + 2
=> a = \(\frac{7b + 1}{3}\)

Only those values of b which will also give a as integer will give us the common values of x
b = 2, 5, 8, 11, 14,...
But for b = 14, we will get x = 7b + 2 = 7*14 + 2 = 100 which is NOT less than 100

=> 4 values of x less than 100 are possible

So, Answer will be 4
Hope it helps!

Watch the following video to learn the Basics of Remainders

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Re: QOTD#14 When x is divided by 3, the remainder is 1. When x [#permalink]
20 Oct 2022, 08:36
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