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Re: If n is an integer, what is the remainder when n is divided
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31 May 2009, 18:02
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vcbabu wrote:
If n is an integer, what is the remainder when n is divided by 7?
1) n+1 is divisible by 7 2) n+8 is divisible by 7
D. Reminder(s) can never be negative but is(are) always: 0 <= r =< 7. Lets say n = ax + r, where a = 7, x is quotient, and r is reminder. Or, n = 7x + r Then in each case above, r = 6.
1) If n = 7x + r, n+1 = (7x+r) + 1. If so, r must be 6. Suff............
If "n= 7x + r" is -ve, x has to be -ve. Then n +1 = (7x + r) + 1 If suppose x = -1, n+1 = 7(-1) + r + 1 = -6+r. What has to be r to have (n+1) divisible by 7? r = +6. Somebody might say -1 but remember r can never be -ve. So what is the minimum r can be 6 because r must be >0 but smaller than 7.
2) If n = 7x + r, n+8 = (7x+r) + 8. Or, n+8 = 7(x+1) + r +1 Now the equation is similar to the eq. in 1. Therefore r = 6 again. Suff................
Re: If n is an integer, what is the remainder when n is divided
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03 Jun 2009, 05:28
Wikipedia has an interesting article on remainder. https://en.wikipedia.org/wiki/Remainder It says Remainder can have negative values as well, unless specified by the condition x < r ≤ x+|d| (or x ≤ r < x+|d|), where x is a constant, d is the divisor, and r is the remainder.
Re: If n is an integer, what is the remainder when n is divided
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21 Jun 2009, 10:40
Expert Reply
goldeneagle94 wrote:
Wikipedia has an interesting article on remainder. https://en.wikipedia.org/wiki/Remainder It says Remainder can have negative values as well, unless specified by the condition x < r ≤ x+|d| (or x ≤ r < x+|d|), where x is a constant, d is the divisor, and r is the remainder.
Several points here:
-First, as that wikipedia article makes clear, mathematicians do not allow remainders to be negative (see where they say "as is usual for mathematicians", when giving the positive solution). ;
-So, when you divide an integer by 7, there are only seven possible remainders: 0, 1, 2, 3, 4, 5 and 6. Remainders can't be negative, they can't be decimals, and they can't be greater than or equal to what you're dividing by;
-I have never seen a real GMAT question that asks about remainders when dividing by a negative number. Nor have I seen questions which ask for the remainder when dividing a negative by a positive number;
-Still, we can see how to find the remainder when dividing, say, -15 by 7. First, we find the remainder when dividing, for example, 20 by 7 as follows:
--we find the nearest multiple of 7 which is *smaller* than 20. That's 14; --we subtract: 20 - 14 = 6. That is, 20 is 6 larger than the nearest smaller multiple of 7, so the remainder is 6 when we divide 20 by 7;
-Doing the same for -15:
--the nearest multiple of 7 which is *smaller* than -15 is -21; --now, -15 - (-21) = 6. That is, -15 is 6 greater than the nearest multiple of 7 which is smaller than -15, and the remainder is 6 when -15 is divided by 7.
So, going back to the original question in this post, the remainder is the same regardless of whether n is positive or negative, and the answer is D.
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