Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 08 Jan 2018
Posts: 98
Location: India
GPA: 4
WE: Information Technology (Computer Software)

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 08:29
Hi, To find the unit digits, we need to consider only the unit of the base and last two digits of power. Another rule is power raised to certain number work in a cycle that means after some powers it will repeat the same digits. Thus, \({97^{275}}\) = \({7^{75}}\) using cycle formula => \({a^{4k+1}}\) = \({a^1}\), where a is base and k is some constant: \({7^{75}}\) = \({7^{3}}\) => 3 (as 75 = 4*18 + 3)
For, \({2^{44}}\) => \({2^4}\) => 6 (as \({a^{4k}}\) = \({a^4}\) and 44 = 4*11)
So, as per question : 36 = 7 (136) so unit digit is 7.
Please hit kudos if you like the solution.



Manager
Joined: 08 Apr 2019
Posts: 157
Location: India
GPA: 4

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 08:29
This question tests your knowledge of the concept of cyclicity.
Since we're only concerned with the unit's place, it's best to rewrite this as 7^275  2^44 (since the tens digit would not have any role to play in determining the units place)
Now, 275 = 4*68 + 3 can be written in the form 4k + 3, and similarly, 44 = 4*11 can be written as 4n
Now, we know that units digit cyclicity of both 7 and 2 is 4, i.e. they repeat their units digit after 4. Knowing this, we get the units digit for 7^(4k+3) to be 3 (eg. 7,49,343) and units digit for 2^(4n) to be 6 (2,4,8,16)
Now, subtracting 6 from 3, we get 7 as the units digit and that's our answer (D)



Intern
Joined: 15 Jun 2019
Posts: 32
Location: Kazakhstan
GPA: 3.93

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 08:31
The last digit of 97^275 is the same as the last digit in 7^275
7^1=7 7^2=49 7^3=343 7^4=2401 7^5=16807 .... So for 7^275 the unit digit is 7
The last digit of 32^44 is the same as the last digit in 2^44 2^1=2 2^2=4 2^3=8 2^4=16 2^5=32
44=4*11 so the unit digit of 2^44 is 6
76=1 Answer: A



Senior Manager
Joined: 11 Feb 2013
Posts: 263
Location: United States (TX)
GMAT 1: 490 Q44 V15 GMAT 2: 690 Q47 V38
GPA: 3.05
WE: Analyst (Commercial Banking)

What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
Updated on: 02 Jul 2019, 09:29
(1) All UNIT DIGITS FOLLOW "THE CYCLICITY OF 4" i.e. after every four powers, UNIT DIGIT REMAINS SAME. So, Divide the all POWERs by 4 and work with the REMAINDER. (IF the REMAINDER is ZERO, take 4 as the remaining number because you have divided the numbers by 4)
A SHORTCUT regarding the DIVISION OF 4: JUST take LAST TWO DIGIT & DIVIDE them by 4. For example, 275/4 is same as 75/4 (REMAINDER=3).
(2) When you are asked to find out UNIT DIGIT, work with UNIT DIGIT only (CROSS OUT TENS & HUNDREDS). For example, UNIT DIGIT of (97^275) and UNIT DIGIT of (9^275) are the SAME. considering cyclic of 4 & unit digit only, the question {what is the unit digit of( 97^3)(32^4) becomes what is the unit digit of (7^3)(2^4)? Here, UNIT DIGIT of 7^3=3 and UNIT DIGIT of 2^4=6 [NOTE: After dividing 44 by 4, REMAINDER is ZERO, for UNIT DIGIT CYCLICITY PURPOSE we will take 4 as remaining number because remainder must be an integer between 1&4).
NOW, CHECK whether (97^275) IS GREATER THAN (32^44)? Case 1: if (97^275) IS GREATER THAN (32^44)?, the value of (97^275) (32^44)=*****************3************6= ****************7 (because we consider 3 as 13 because first term is GREATER) Case 1: if (97^275) IS LESS THAN (32^44)?, the value of (97^275) (32^44)=*****************3************6= **********3 (because simply 6 MINUS 3 because SECOND term is GREATER)
CHECKING: 97>32 & 275>44. SO, (97^275) IS DEFINITELY GREATER THAN(32^44). SO, ONLY CASE 1 POSSIBLE.
so, UNIT DIGIT=7 (D is the ANSWER)



Manager
Joined: 21 Jan 2019
Posts: 102

What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 08:32
Quote: What is the digit on the units place in the expanded value of 97275–324497275–3244?
A. 1 B. 3 C. 5 D. 7 E. 9
in no. 97, cyclicity of unit's digit 7 is 7,9,3,1 and 275 /4 gives 3 as a remainder. in no.32, cyclicity of unit's digit 2 is 2,4,8,6 and 44/4 gives 0 as remainder. hence the equation is ...3...6 which will end up to be 7 as the unit's digit. hence option D



Manager
Joined: 23 Oct 2018
Posts: 50

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 08:34
What is the digit on the units place in the expanded value of 97^275–32^44?
Since we are only asked about the unit digit of the above expression, no. cyclicity principle will help.
7 has a cyclicity of 4 3,9,7,1 divide 275 by 4 and we get the remainder 3 so the unit digit of the expression 97^275 will be 3. Same way, 2 has a cyclicity of 4 2,4,6,8 divide 44 by 4 and there is no remainder, so the unit digit is 6.
Try some nos. 136 will have unit digit 7, 236 will have unit digit 7 and so on.



Manager
Joined: 01 Oct 2018
Posts: 112

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 08:37
(97^275)  32^44 We interested in only the last digit, so: 7^275  2^44 7^1 = 7 7^2 = 9 7^3 = 3 7^4 = 1 7^5 = 7
So this pattern repeat, that's mean: 272 is the last multiple of 4 before 275 Number273274275 LastDigit793 Ok, last digit of 7^275 is 3
Than make the same analyze with 2^44 last digit is 6
So 3  6 = 3 this can't becase (97^275) > 32^44, so 13  6 = 7
Answ D
Posted from my mobile device



Intern
Joined: 15 Sep 2017
Posts: 44
Location: United States (IL)
GPA: 3.4

What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
Updated on: 02 Jul 2019, 11:13
Answer DTo find the unit digit of the given number we need to know the cyclicity of 7 and 2 and that is 4 numbers after that it repeats the same numbers. 7: 7931 2 : 2486 Therefore unit digit of 97^275 is 3 and unit digit of 32^44 is 6 36 = 7Answer is D
_________________
Originally posted by Tashin Azad on 02 Jul 2019, 08:41.
Last edited by Tashin Azad on 02 Jul 2019, 11:13, edited 1 time in total.



Intern
Joined: 21 Feb 2018
Posts: 16

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 08:44
97^275  32^44. Lets assume this to be A  B
To find the units digit of a AB, we must first know the unit's digit of A and B respectively.
Unit's Digit of A  97^275 depends on the unit's digit when obtained from 7^275. Since 7 has a cyclicity of 4(7,9,3,1) and 275 = 4(68) + 3 => the unit's digit of A will be 3
Similarly, Unit's Digit of A  32^44 depends on the unit's digit when obtained from 2^44. Since 2 has a cyclicity of 4(2,4,8,6) and 44 = 4(11) => the unit's digit of A will be 6
Unit's digit of AB = 3  6 = 13 (By Borrowing from the Ten's digit in A)  6 = 7
Hence, The answer must be D



Manager
Joined: 12 Mar 2018
Posts: 83
Location: United States

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 08:45
The units digit of 97^275 will be 3 and the units digit of 2^44 will be 6. So the units digit of the difference then will be 7.



Manager
Joined: 24 Jun 2019
Posts: 113

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 08:48
97^275: Use only last digit to get the units place:
Power1  7^1 = 7 Power2  7x7 = 49 Power3  9x7 = 63 Power4  3x7= 21 Power5  1x7 = 7 .... same as 1
so units digit will cycle through 7, 9, 3, 1... four unique digits repeating
275 divde by 4 gives quotient 272 and remainder 3  so units digit will be 3 (3rd in the cycle)
32^44: Same logic as above Power1  2^1 = 2 Power2  2x2 = 4 Power3  4x2 = 8 Power4  8x2 = 16 Power5  6x2 = 12 .... same units digit as 1
so the cycle here is 2, 4, 8, 6.... again cycle of 4 digits
44 is divisible by 4  so 44th power of 2 will have units digit 6 (4th in cycle)
Units digit of difference will be 3  6 = 136 (Do manual subtraction on paper  1 will be carried to 3 to make it 13) = 7
Ans is D  7



Senior Manager
Joined: 28 Feb 2014
Posts: 270
Location: India
Concentration: General Management, International Business
GPA: 3.97
WE: Engineering (Education)

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 08:52
What is the digit on the units place in the expanded value of 97^275–32^44?
A. 1 B. 3 C. 5 D. 7 E. 9
This can be done with the help of cyclicity of 7 and 2 which is 4. Question can be rephrased as what is the unit digit of 7^275  2^44 on dividing 275 with 4 (cyclicity of 7) we get remainder as 3 and dividing 44 with 4 (cyclicity of 2) we get remainder as 0 which will be equal to 4 itself Unit digit of 7^3  2^4 = 3  6 = 7 (at one's place)



Manager
Joined: 18 Apr 2019
Posts: 87
Location: India
GPA: 4

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 08:54
Concept tested: This question is based on cyclicity. The cyclicity of 7 is 7,9,3,1 and that of 2 is 2,4,8,6. Soln: Now, if we only see the units digit and the power. 7 is raised to 275. 275 when divided by 4 has quotient 68 and leaves remainder 3. What is means is 7 completes 68 cycles of 7,9,1,3 and then 3 units are left. So taking the 3rd digit in cyclicity, we have the units digit of 37^275 as 3. Applying the same concept to 2, we come to the conclusion that it completes 11 cycles and leaves no remainder. Hence units digit is the 4th digit in the cyclicity  6. Now just consider the first 2 digit number that ends with 3 and subtract 6 from it. This gives you the answer 7. [D] Note: we can't subtract 3 from 6 and say the remainder is 3. So we take a number that is greater than 6 and ends with 3.



Senior Manager
Joined: 31 May 2018
Posts: 461
Location: United States
Concentration: Finance, Marketing

What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
Updated on: 02 Jul 2019, 08:57
we need to find the unit digit of 97^275  32^44
so we will consider unit digits of both \(7^1\)=7 \(7^2\)=9 (unit digit) \(7^3\)=3 \(7^4\)=1 \(7^5\)=7 \(7^6\)=9 \(7^7\)=3 \(7^8\)=1 from here we conclude that it follows a cyclic pattern \(7^4\),\(7^8\) = each unit digit = 1 we need to find unit digit of 7^275 so we will write this in terms of \(7^4\)  (7^4)^68 * \(7^3\) = 1*\(7^3\) = 3 (unit digit of 97^275)
now unit digit of 2^44 we need to find cyclic pattern by performing the same operation above on 2 we find pattern \(2^4\),\(2^8\) = each unit digit = 6 so we will write 2^44 in terms of \(2^4\) (2^4)^11 = 6^11 = 6 (unit digit of 32^44)
the difference of unit digit 97^275  32^44
(...........3)  (....6) = 7 (since 97^275 is larger value than 32^44)
correct answer is 7 option D
Originally posted by shridhar786 on 02 Jul 2019, 08:55.
Last edited by shridhar786 on 02 Jul 2019, 08:57, edited 1 time in total.



Manager
Joined: 30 May 2019
Posts: 109

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 08:56
Here, concept of cyclicity is tested. They gave us these huge, ugly looking numbers to distract and make us panic. But we won't. For 97^275 it is enough to know units digit of 7^275. Cyclicity of 7 is 4. that is 7^1=07 7^2=49 7^3=_43 7^4=__01 So, units digit when 7 is raised to the power of 275 is 43. Likewise, we need to know remainder of 2^44. 2 has also cyclicity of 4, that is 2^1=02 2^2=04 2^3=08 2^4=16, So, units digit when 2 is raised to the power of 44 is 6, So 436=_7 (D)



Intern
Joined: 08 Nov 2016
Posts: 19
Location: India
GPA: 3.99
WE: Web Development (Computer Software)

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 09:01
What is the digit on the units place in the expanded value of 97^275–32^44?
A. 1 B. 3 C. 5 D. 7 E. 9
We can get the series of unit places for all powers of 7 & 2 to see the series of repetition. 7^1 = unit digit 7 7^2 = unit digit 9 7^3 = unit digit 3 7^4 = unit digit 1 7^5 = unit digit 7 , So after 4 it gets repeated. Same with 2, after 4, unit digits get repeated.
Now if we calculate for power 275 for 7, 3 is remainder and the unit digit should be "3", and for power 44 for 2, remainder is 0, and unit digit will be "6". Now unit digit(3)  unit digit(6) = 136 = 7. Answer is 7.



Intern
Joined: 23 Jul 2017
Posts: 20
Location: India
Concentration: Technology, Entrepreneurship
GPA: 2.16
WE: Other (Other)

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 09:05
Any power of a number ending with 7 will have the units digit either 7,9,3 or 1 and any number ending with 2 will have the units digit either 2,4,8 or 6. Therefore 97^275 and 32^44 will be ending with 3 and 6, giving the answer as 7.



Intern
Joined: 09 Feb 2019
Posts: 20

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 09:07
7th & 2nd unit place repeats after every 4 multiple
7^1=7 7^2=4 7^3=3 7^4=1 7^5=7
2^1=2 2^2=4 2^3=8 2^4=6 2^5=2
So the answer is 76 =1 (A)



Manager
Joined: 15 Nov 2015
Posts: 158
Location: India
GPA: 3.7

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 09:07
Both 7&2 has power cycle of 4
Units digit of 36 =7
Hence option D



Intern
Joined: 04 Feb 2019
Posts: 5

Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
Show Tags
02 Jul 2019, 09:09
This can be simplified as units digit of 7^275  2^44
We know periodicity of 7 is 4 ie. (7^4)^n will return 1 in units digit for all n>=1. Now 275 = 4 x 68 + 3, so the units digit is determined by units digit of 7^3 ie. 3.
Again we know We know periodicity of 2 is 4 ie. (2^4)^n will return 6 in units digit for all n>=1. Now 44 = 4 x 11, so the units digit always 6.
Units digit 3  Units digit 6 = Units digit 7
Answer is D




Re: What is the digit on the units place in the expanded value of 97^275 –
[#permalink]
02 Jul 2019, 09:09



Go to page
Previous
1 2 3 4 5 6
Next
[ 107 posts ]



