FROM Veritas Prep Blog: A Tricky Question on Negative Remainders

Today, we will discuss the question we left you with last week. It involves a lot of different concepts – remainder on division by 5, cyclicity and negative remainders. Since we did not get any replies with the solution, we are assuming that it turned out to be a little hard. It actually is a little harder than your standard GMAT questions but the point is that it can be easily solved using all concepts relevant to GMAT. Hence it certainly makes sense to understand how to solve it.   Question: What is the remainder when 3^(7^11) is divided by 5? (here, 3 is raised to the power (7^11)) (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 Solution: As we said last week, this question can easily be solved using cyclicity and negative remainders. What is the remainder when a number is divided by 5? Say, what is the remainder when 2387646 is divided by 5? Are you going to do this division to find the remainder? No! Note that every number ending in 5 or 0 is divisible by 5. 2387646 = 2387645 + 1 i.e. the given number is 1 more than a multiple of 5. Obviously then, when the number is divided by 5, the remainder will be 1. Hence the last digit of a number decides what the remainder is when the number is divided by 5. On the same lines, What is the remainder when 36793 is divided by 5? It is 3 (since it is 3 more than 36790 – a multiple of 5). What is the remainder when 46^8 is divided by 5? It is 1. Why? Because 46 to any power will always end with 6 so it will always be 1 more than a multiple of 5. On the same lines, if we can find the last digit of 3^(7^11), we will be able to find the remainder when it is divided by 5. Recall from the discussion in your books, 3 has a cyclicity of 4 i.e. the last digit of 3 to any power takes one of 4 values in succession. 3^1 = 3 3^2 = 9 3^3 = 27 3^4 = 81 3^5 = 243 3^6 = 729 and so on… The last digits of powers of 3 are 3, 9, 7, 1, 3, 9, 7, 1 … Every time the power is a multiple of 4, the last digit is 1. If it is 1 more than a multiple of 4, the last digit is 3. If it is 2 more than a multiple of 4, the last digit is 9 and if it 3 more than a multiple of 4, the last digit is 7. What about the power here 7^(11)? Is it a multiple of 4, 1 more than a multiple of 4, 2 more than a multiple of 4 or 3 more than a multiple of 4? We need to find the remainder when 7^(11) is divided by 4 to know that. Do you remember the binomial theorem concept we discussed many weeks back? If no, check it out here. 7^(11) = (8 – 1)^(11) When this is divided by 4, the remainder will be the last term of this expansion which will be (-1)^11. A remainder of -1 means a positive remainder of 3 (if you are not sure why this is so, check last week’s post here). Mind you, you are not to mark the answer as (D) here and move on! The solution is not complete yet. 3 is just the remainder when 7^(11) is divided by 4. So 7^(11) is 3 more than a multiple of 4. Review what we just discussed above: If the power of 3 is 3 more than a multiple of 4, the last digit of 3^(power) will be 7. So the last digit of 3^(7^11) is 7. If the last digit of a number is 7, when it is divided by 5, the remainder will be 2. Now we got the answer! Answer (C) Interesting question, isn’t it? Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

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hi ...
another fast way to such Q is through remainder theorem(?)..
we find modulus and here it is 5*(1-1/5)=4..
this means the remainder asked will be same as remainder of 3^(remainder of 7^11 when divided by 4) divided by 5..
7^11 when divided by 4 will give remainder of (-1)^11=-1 or 3...
so 3^3 divided will give 27/5 or 2 is the remainder...
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When getting the pattern of the units digit of x^k you should always start with k=1, not k=0.
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The red part is not correct: 3^7^11 = 3^77 but 3^(7^11) = 3 to the power of (7^11).

Theory on Exponents: math-number-theory-88376.html
Tips on Exponents: exponents-and-roots-on-the-gmat-tips-and-hints-174993.html

Check Units digits, exponents, remainders problems directory in our Special Questions Directory.
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When you divide 3^k by 5 or 10, and want to know the remainder, then you want to know where you'll be in the units digit pattern of 3^k. Because the units digit pattern of 3^k repeats in 'blocks' (cycles) of 4 terms, you need to know the remainder when you divide k by 4 in order to know where you'll be in that pattern. In this question, k = 7^11, so you'd need to find the remainder when you divide 7^11 by 4. What you found above is the remainder when you divide 7^11 by 10. So you correctly determined that 7^11 has a units digit of 3, but that alone doesn't tell you the remainder when you divide 7^11 by 4 (since some numbers that end in 3 will give a remainder of 1 when you divide by 4, and some will give a remainder of 3 when you divide by 4). It's because it's not straightforward to find the remainder when you divide 7^11 by 4 that the posters above needed some lengthier solutions.
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