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# Veritas Prep Blog

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Veritas Prep Representative
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Veritas Prep Representative
Joined: 21 Jan 2010
Posts: 99
Own Kudos [?]: 654 [0]
Given Kudos: 2
Veritas Prep Representative
Joined: 21 Jan 2010
Posts: 99
Own Kudos [?]: 654 [0]
Given Kudos: 2
Veritas Prep Representative
Joined: 21 Jan 2010
Posts: 99
Own Kudos [?]: 654 [1]
Given Kudos: 2
A Tricky Question on Negative Remainders [#permalink]
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 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) 4Solution: 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 + 1i.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 = 33^2 = 93^3 = 273^4 = 813^5 = 2433^6 = 729and 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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Veritas Prep Representative
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School Profile: The Path of Enrichment at Williams College [#permalink]
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GMAT at the Movies: What Austin Powers Can Teach You about S [#permalink]
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How Hard is the Verbal Section of the GMAT? [#permalink]
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GMAT Tip of the Week: The Whole Sentence Mathers [#permalink]
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Looking for Similar Triangles on the GMAT [#permalink]
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School Profile: International Exploration, Tenting, and Pizz [#permalink]
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The Most Efficient Way to Study Least Common Multiples on th [#permalink]
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GMAT Tip of the Week: ASAP Test Taking Can Be Rocky (That's [#permalink]
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Determining the Area of Similar Triangles on the GMAT [#permalink]
 FROM Veritas Prep Blog: Determining the Area of Similar Triangles on the GMAT Recall the important property that we discussed about the relation between the areas of the two similar triangles last week – if the ratio of their sides is ‘k’, the ratio of their areas will be k^2. As mentioned last week, it’s an important property and helps you easily solve otherwise difficult questions. The question I have in mind today also brings in focus the Pythagorean triplets.There are some triplets that you should know out cold: (3, 4, 5), (5, 12, 13) and (8, 15, 17). Usually you will find one of these three or their multiples on GMAT. Given a right triangle and two sides, say the two legs, of length 20 and 48, we need to immediately bring them down to the lowest form 20:48 = 5:12. So we know that we are talking about the 5, 12, 13 triplet and the hypotenuse will be 13*4 = 52. These little things help us save a lot of time. Why is it that some people get done with the Quant section in less than an hour while others fall short on time? It is these little things that an adept test taker has mastered which make all the difference.Anyway, let us go on to the question we have in mind.Question: In the figure given below, the length of PQ is 12 and the length of PR is 15. The area of right triangle STU is equal to the area of the shaded region. If the ratio of the length of ST to the length of TU is equal to the ratio of the length of PQ to the length of QR, what is the length of TU?(A) (9√2)/4(B) 9/2(C) (9√2)/2(D) 6√2(E) 12Solution: The information given in the question seems to overwhelm us but let’s take it a bit at a time.“length of PQ is 12 and the length of PR is 15”PQR is a right triangle such that PQ = 12 and PR = 15. So PQ:PR = 4:5. Recall the 3-4-5 triplet. A multiple triplet of 3-4-5 is 9-12-15. This means QR = 9.“ratio of the length of ST to the length of TU is equal to the ratio of the length of PQ to the length of QR”ST/TU = PQ/QRThe ratio of two sides of PQR is equal to the ratio of two sides of STU and the included angle between the sides is same ( = 90). Using SAS, triangles PQR and STU are similar.“The area of right triangle STU is equal to the area of the shaded region”Area of triangle PQR = Area of triangle STU + Area of Shaded RegionSince area of triangle STU = area of shaded region, (area of triangle PQR) = 2*(area of triangle STU)In similar triangles, if the sides are in the ratio k, the areas of the triangles are in the ratio k^2. If the ratio of the areas is given as 2 (i.e. k^2 is 2), the sides must be in the ratio √2 (i.e. k must be √2).Since QR = 9, TU must be 9/√2. But there is no 9/√2 in the options – in the options the denominators are rationalized. TU = 9/√2 = (9*√2)/(√2*√2) = (9*√2)/2.Answer (C)The question could take a long time if we do not remember the Pythagorean triplets and the area of similar triangles property.Takeaways:Pythagorean triplets you should know: (3, 4, 5), (5, 12, 13) and (8, 15, 17) and their multiples.In similar triangles, if the sides are in the ratio k, the areas of the triangles are in the ratio k^2.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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Can I Get Accepted into Harvard, Stanford, or Wharton with a [#permalink]
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