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If a is a positive integer, and if the units digit of a^2 is [#permalink]
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10 Feb 2011, 15:56
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If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2? A. 1 B. 3 C. 5 D. 6 C. 14 I guess I am just not reading this properly. I don't understand what they mean by units digit...
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If a is a positive integer, and if the units digit of a^2 is [#permalink]
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ChenggongMAS wrote: If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2?
A) 1 B) 3 C) 5 D) 6 C) 14
I guess I am just not reading this properly. I don't understand what they mean by units digit... 1234.567 1  THOUSANDS 2  HUNDREDS 3  TENS 4  UNITS .  decimal point 5  TENTHS 6  HUNDREDTHS 7  THOUSANDTHS So thE units digit is the digit to the left of the decimal point or in integer it's the rightmost digit. For example: the units digit of 1.2 is 1 and the units digit of 13 is 3. Back to the original question. If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2?The units digit of a^2 is 9 > the units digit of a itself is either 3 or 7 (3^2= 9 and 7^2=4 9); The units digit of (a+1)^2 is 4 > the units digit of a+1 is either 2 or 8 (2^2= 4 and 8^2=6 4), so the the units digit of a itself is either 21=1 or 81=7; To satisfy both conditions the units digit of a must be 7. Now, a+2 will have the units digit equal to 7+2=9, thus the units digit of (a+2)^2, will be 1 (9^2=8 1). Answer: A. Check Number Theory chapter of Math Book for more: mathnumbertheory88376.html
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Re: Properties of Numbers [#permalink]
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10 Feb 2011, 16:33
Thanks Bunuel. It is more clear now.



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Re: If a is a positive integer, and if the units digit of a^2 is [#permalink]
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19 Apr 2012, 05:40
ChenggongMAS wrote: If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2?
A. 1 B. 3 C. 5 D. 6 C. 14
I guess I am just not reading this properly. I don't understand what they mean by units digit... Substitution Method was followed to get the Answer A: 1 guess values: 3 satisfies 2nd condition, but not 3rd condition... but 7 satisfied both conditions and hence the answer 1 was obatined due the sq of 9
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Re: Properties of Numbers [#permalink]
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06 Feb 2013, 19:35
Bunuel wrote: ChenggongMAS wrote: If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2?
A) 1 B) 3 C) 5 D) 6 C) 14
I guess I am just not reading this properly. I don't understand what they mean by units digit... 1234.567 1  THOUSANDS 2  HUNDREDS 3  TENS 4  UNITS .  decimal point 5  TENTHS 6  HUNDREDTHS 7  THOUSANDTHS So th units digit is the digit to the left of the decimal point or in integer it's the rightmost digit. For example: the units digit of 1.2 is 1 and the units digit of 13 is 3. Back to the original question. If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2?The units digit of a^2 is 9 > the units digit of a itself is either 3 or 7 (3^2= 9 and 7^2=4 9); The units digit of (a+1)^2 is 4 > the units digit of a+1 is either 2 or 8 (2^2= 4 and 8^2=6 4), so the the units digit of a itself is either 21=1 or 81=7; To satisfy both conditions the units digit of a must be 7. Now, a+2 will have the units digit equal to 7+2=9, thus the units digit of (a+2)^2, will be 1 (9^2=8 1). Answer: A. Check Number Theory chapter of Math Book for more: mathnumbertheory88376.htmlI was wondering if there's any algebraic soln to this question.
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Re: If a is a positive integer, and if the units digit of a^2 is [#permalink]
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08 Feb 2013, 09:40
ChenggongMAS wrote: If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2?
A. 1 B. 3 C. 5 D. 6 C. 14
I guess I am just not reading this properly. I don't understand what they mean by units digit... For unit digit of a^2 to be 9...unit digit of a has to be 3 or 7... Now for unit digit of (a+1)^2 to be 4..unit digit of a has to be 1 or 7.... From the above two conditions, unit value of a has to be 7, which will satisfy both the conditions... Now id unit digit of a is 7, unit digit of (a+2)^2 hast to be 1..



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Re: Properties of Numbers [#permalink]
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08 Feb 2013, 13:36
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Sachin9 wrote: Bunuel wrote: I was wondering if there's any algebraic soln to this question.
1. Units digit of a^2 is 9. 2. (a+1)^2 = a^2 + 2a + 1 .....UD of (a^2) + UD of (2a) + 1 = 4 ....9+1 +UD(2a)=4 .....10+UD(2a) = 4...therefore, a = 2 or 7 Based on 1 and 2, a can't be 2, so it has to be 7. We can calculate (a+2)^2 (a+2)^2 = a^2 + 4a + 4 = a^2 + 2(2a) +4 = UD(a^2)+2(UD of 2a) +4 ..this gives you units digit of 1...and thats the answer.



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Re: If a is a positive integer, and if the units digit of a^2 is [#permalink]
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Re: If a is a positive integer, and if the units digit of a^2 is [#permalink]
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12 May 2014, 03:05
Hi Bunuel, This one too is tagged as 'hard' in GMATPrep. While it is marked as sub 600 here. Thanks!
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Re: If a is a positive integer, and if the units digit of a^2 is [#permalink]
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12 May 2014, 03:14



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Re: If a is a positive integer, and if the units digit of a^2 is [#permalink]
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12 May 2014, 03:33
Bunuel, Thanks for your reply. Yup, it does! However, I have not seen this logic hold true in every case. What are the percentage ranges for sub600, 600700, +700 etc? This will help me point out incorrect tags if any so as to improve this forum. And quite frankly I didnt find this question that easy. But cannot argue against the statistics unless those 101 users somehow were not representative of an average test taker. Moreover, I have heard GMAC also categorizes questions based on how many test takers got it right/wrong. With hundreds of thousands taking the gmat each year they are likely to have bigger data. I am just trying to understand tagging here. Thanks for your understanding.
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Re: If a is a positive integer, and if the units digit of a^2 is [#permalink]
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12 May 2014, 03:40
MensaNumber wrote: Bunuel, Thanks for your reply. Yup, it does!
However, I have not seen this logic hold true in every case. What are the percentage ranges for sub600, 600700, +700 etc? This will help me point out incorrect tags if any so as to improve this forum.
And quite frankly I didnt find this question that easy. But cannot argue against the statistics unless those 101 users somehow were not representative of an average test taker. Moreover, I have hard GMAC also categorizes questions based on how many test takers got it right/wrong. With hundreds of thousands taking the gmat each year they are likely to have bigger data.
I am just trying to understand tagging here. Thanks for your understanding. Well, you can judge the difficulty level of a question based on the statistics and not on the tags. I agree that GMAC has larger data and their stats might be more representative. Having said that I must add that still the difficulty level is quite subjective issue.
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Re: If a is a positive integer, and if the units digit of a^2 is [#permalink]
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12 May 2014, 03:49
Yes difficulty is a subjective matter. Hence I think defining percentage ranges corresponding to sub600, 600700 and +700 is a great way to bring in objectivity? Or do we have these ranges already? Thanks!
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Re: If a is a positive integer, and if the units digit of a^2 is [#permalink]
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Re: If a is a positive integer, and if the units digit of a^2 is [#permalink]
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13 May 2014, 01:44
Bunuel wrote: % of incorrect answers  Difficulty 0  29 = low (sub600) 30  69 = medium (600700) 70  99 = hard (700+) Great! Will follow those ranges. Thanks
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Re: If a is a positive integer, and if the units digit of a^2 is [#permalink]
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Re: If a is a positive integer, and if the units digit of a^2 is [#permalink]
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07 Jul 2015, 21:28
kapsycumm wrote: Sachin9 wrote: Bunuel wrote: I was wondering if there's any algebraic soln to this question.
1. Units digit of a^2 is 9. 2. (a+1)^2 = a^2 + 2a + 1 .....UD of (a^2) + UD of (2a) + 1 = 4 ....9+1 +UD(2a)=4 .....10+UD(2a) = 4...therefore, a = 2 or 7 Based on 1 and 2, a can't be 2, so it has to be 7. We can calculate (a+2)^2 (a+2)^2 = a^2 + 4a + 4 = a^2 + 2(2a) +4 = UD(a^2)+2(UD of 2a) +4 ..this gives you units digit of 1...and thats the answer. We have been given a^2 = 9. that means a = +/ 3. We put +3 in (a+1)^2, it doesn't give us 4. But if we put 3, (3+1)^2 = (2)^2 = 4. Similarly, in (a+2)^2 = (3+2)^2 = (1)^2 = 1. Hence, the answer is 1. Or using quadratic equation. (a+1)^2 = 4 a^2 + 2a(1) + 1^2 = 4 We know a^2 = 9. 2a = 491 2a = 6 a = 3. Substitute, this in (a+2)^2. We get 1. Let me know if it makes sense or my logic is flawed.



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Re: If a is a positive integer, and if the units digit of a^2 is [#permalink]
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20 Aug 2015, 02:51
Hanish Satija wrote: We have been given a^2 = 9. that means a = +/ 3.
Yes there is a flaw...a is a positive integer...it cannot be 3



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If a is a positive integer, and if the units digit of a^2 is [#permalink]
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14 Jul 2016, 09:55
Gentlemen, Good afternoon. It´s my first time in the Forum  I am glad to see such a nice resource! Question: How can I be sure that if the units digit of (a^2 ) = 9 , for sure the units digit of "a" must be 3 or 7 ? I have followed the answer by expanding the equations and adding the units digits, which I did too, but took quite a longer time. My first thought when I saw the quation was this " units digit of "a" must e 7 or 9 " approach, however it just sounded in my mind like good a guess  how can I be sure that no other number squared from 0 to infinite will result in a number with 9 ,( or x, or y) in the units digit ? What theory am I missing, guys? Thank you and luck to all!



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Re: If a is a positive integer, and if the units digit of a^2 is [#permalink]
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14 Jul 2016, 10:15
itabra wrote: Gentlemen, Good afternoon. It´s my first time in the Forum  I am glad to see such a nice resource! Question: How can I be sure that if the units digit of (a^2 ) = 9 , for sure the units digit of "a" must be 3 or 7 ? I have followed the answer by expanding the equations and adding the units digits, which I did too, but took quite a longer time. My first thought when I saw the quation was this " units digit of "a" must e 7 or 9 " approach, however it just sounded in my mind like good a guess  how can I be sure that no other number squared from 0 to infinite will result in a number with 9 ,( or x, or y) in the units digit ? What theory am I missing, guys? Thank you and luck to all! If x is an integer to get the units digit of x^2 the only thing we need to know is the units digit of x itself. There are ten digits, so we can have only the following cases: 0^2 = 01^2 = 12^2 = 43^2 = 94^2 = 1 65^2 = 2 56^2 = 3 67^2 = 4 98^2 = 6 49^2 = 8 1As you can see only if an integer ends with 3 or 9 its square will have the units digit of 9.
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