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What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 08:00
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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
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Re: What is the digit on the units place in the expanded value of 97^275 –
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03 Jul 2019, 06:34
We are looking at the unit digits so we just need to consider the unit digit of each number. So we need to look at 7^275  2^44 Now, unit digits of 7^1 = 7 7^2 = 9 7^3 = 3 7^4 = 1 7^5 = 7 So, we see after every 4th power the unit digit starts repeating. We essentially need 7^3 as 275 = 4*68 + 3 Similarly 44 is a multiple of 4 so we will look at unit digit of 2^4 The problem is reduced to 3  6. As 3 is a small number than 6 and is present at the units place, it will take a carry over from the tens place. So, it becomes 13  6 = 7



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Re: What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 14:05
Any number ending in 7 when raised to a power will have the following pattern 7,9,3,1 as the units digit and Any number ending in 2 when raised to a power will have the following pattern 2,4,8,6 as the units digit Now 97^275 means we divide 275 by 4 and compare it against the pattern. 275th power will have 3 as the units digit and Now 32^44 means we divide 44 by 4 and compare it against the pattern. 44th power will have 6 as the units digit Thus we have 3  6 >>> The trick is that you have to imagine the normal subtraction and get 1 as the carry over. thus it is actually 13  6 Answer is 7
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What is the digit on the units place in the expanded value of 97^275 –
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Updated on: 02 Jul 2019, 09:39
Cyclicity of 7 is 4 so \(97^{275}\) has a units digit same as that of 7 raised to the remainder of \(\frac{275}{4}\) which is \(7^3\)=343
Therefore \(97^{275}\) has a units digit of \(3\)
Similarly \(32^{44}\) has a units digit \(6\) since cyclicity of \(2\) is \(4\)
Therefore the difference is xxxxxxx3  yyyyyyy6 = zzzzz7
Answer is D
Originally posted by firas92 on 02 Jul 2019, 08:06.
Last edited by firas92 on 02 Jul 2019, 09:39, edited 5 times in total.



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What is the digit on the units place in the expanded value of 97^275 –
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Updated on: 03 Jul 2019, 13:28
for \(97^{275}\), when 275 is divided by 4, remains 3 , so the unit digit is similar to that of \(7^3\) which is 3 for \(32^{44}\), when 44 is divided by 4, remains 0 , so the unit digit is similar to that of \(2^4\) which is 6 so the unit digit is (36) = 7
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Re: What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 08:12
We have to find unit digit of 97^275–32^44 275=3 mod 4 Hence unit digit of 97^275 is same as 7^3=3
44=0 mod 4 Hence unit digit of 32^44 is same as 2^4=6
unit digit of 97^275–32^44= x3y6=7



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Re: What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 08:13
The cyclicity of 7 and 2 is 4.That is the last digit repeats after every four powers.
Say for 7 7 9 3 1 again 7 9 3 1 Similarly for 2 it is 2,4,8,6
Now 97^275 implies last digit is 7 ^ 3= 3 (275/4 leaves 3 )
32^44 implies last digit is 6 (44/2 leaves 0 )
Therefore last digit = 36 = 136= 7 (last digit will not be negative we will have to borrow one )



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Re: What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 08:14
Cyclicity of last digit of 7= {7, 9,3,1,7.......} Cyclicity of last digit of 2= {2,4,8,6,2........} so every power of form 4K+1 has same last digit. Refer Image. Ans . 7
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Re: What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 08:15
Cyclicity of 7= 4. Notice the unit's digit 7^1=7, 7^2=9 7^3=3 7^4=1 7^5=7 (repeats) Remainder when 275 divided by 4 = 3 Therefore Unit's digit of 97^275=> 7^3 => 3 Likewise, cyclicity of 2=4 And Remainder when 44 divided by 4 = 0 Therefore, Unit's digit of 32^44 => 2^4 => 6 Difference 36 = 7
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Re: What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 08:18
use cyclicity to solve so 97^275 ; unit digit ; 3 and 32^44; unit digit ; 6 ∆ ; 36 ; 7 IMO D 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
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Re: What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 08:20
Based on cyclic properties of units digit 7's and 2's unit digit repeats itself in ^4. so 7^275 units digit similar to 7^3 and 2^44 will be similar to 2^4. 34316 gives unit digit of 7. IMO D
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Re: What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 08:21
Units place for 7^275 = 3 Units place for 2^44 = 6
Now 36 = 3. We need to add 10 to any negative difference in such questions to get the actual value of units place.
Therefore, answer is D.



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Re: What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 08:21
IMO D
7 has cyclicity of 4, that is 7*7*7*7 has unit's digit as 1 2 has cyclicity of 4, that is 2*2*2*2 has unit's digit as 6
Now we just have to reorder the given expression 97^275–32^44 = ((97)^(4*68)) * (97^3)  32^(4*11) => 7^3 has unit's digit 3  2^(4*11) has unit's digit 6 => answer is 7



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Re: What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 08:22
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 In 97^275, unit digit will depend on 7^275 and in 32^44 will depend on 2^44. Unit digit of 7^1 = 7 7^2 = 9 7^3 = 3 7^4 = 1 7^5 = 7 In general unit digit of 7^(4x)=1 275=4*68+3 Unit digit of 97^275 = 3 Similarily, Unit digit of 2^1 = 2 2^2 = 4 2^3 = 8 2^4 = 6 2^5 = 2 In general unit digit of 2^(4x)=6 44=4*11 Unit digit of 32^44 = 6 Unit digit of 97^27532^44 = 13 6 = 7 since 36 gives negative value and one is borrowed from 10's digit. IMO D
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Re: What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 08:22
Concept : Cyclicity
97^275  32^44 Unit digit of 97 is same as 7 and 32 is same as 2 > 97^275  32^44 > 7^275  2^44 Cyclicity of 7 is 4 > 275 = 4*68 + 3 > 7^275 = 7^3 = 343 Unit Digit = 3
Cyclicity of 2 is 4 > 44 = 4*10 + 4 > 2^44 = 2^4 = 16 Unit Digit = 6
So, Unit Digit of 7^275  2^44 = 3  6 = 13  6 = 7
IMO Option D
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Re: What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 08:22
7^3 ends with 3 2^4 ends with 6 When subtraction it gives 7
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Re: What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 08:23
My answer is (D), i.e. 7. The concept is pretty straightforward, but it can be time consuming. I just need to speed up.
97^275 is clearly larger than 32^44. So after the subtraction, the result is positive. We just need to know the unit digit for both 97^275 and 32^44.
Notice 97 ^ 1 ending in 7 97 ^ 2 ending in 9 97 ^ 3 ending in 3 97 ^ 4 ending in 1 97 ^ 5 ending in 7, which is the same as 97 ^ 1. We already see a pattern here. ... 97 ^ 275 = 97 ^ (4 * 68 + 3) ending in 3.
Similarly, for 32 ^ 44, we can find out it ends in 6.
Because 13  6 = 7, I choose (D)



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Re: What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 08:25
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
Answer: D power>unit digit  7^1 >7 7^2>9 7^3 > 3: 275 = 4* 68 + 3 7^4>1 7^5>7 >> so repeating after 4th power: 79317931
power>unit digit  2^1>2 2^2>4 2^3>8 2^4>6: 44 = 4*11 + 0 2^5>2>> so repeating after 4th power: 24862486
97^275 >32^44, so 36 ==> 7



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What is the digit on the units place in the expanded value of 97^275 –
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Updated on: 02 Jul 2019, 22:26
Think answer is D. 97^275  32^44 7^1=7 7^2=9 7^3=3 7^4=1 7^5=7 .. the trend continues 7,9,3,1 2^1=2 2^2=4 2^3=8 2^4=6 2^5=2 .. the trend continues 2,4,8,6 97^275  32^44 Effectively 7^275  2^44 7^275 will have the same last digit as 7^3 2^44 will have the same last digit as 2^4 7^32^4 3  6 (Same as 13  6 carried over from the tens digfit) = 7 Posted from my mobile device
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Originally posted by prashanths on 02 Jul 2019, 08:25.
Last edited by prashanths on 02 Jul 2019, 22:26, edited 3 times in total.



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Re: What is the digit on the units place in the expanded value of 97^275 –
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02 Jul 2019, 08:26
7 and 2 follow a cycle when it comes to the unit's place. 7^1= 7, 7^2= 49 , 7^3=343, 7^4= 2401, 7^5= 16807. Hence, the unit's digit has a cycle of 7,9,3,1. By finding the remainder when 275 is divided by 4, we can find the corresponding term in the series which gives us a remainder of 3. Hence, the unit's digit would be 3 for the first term. Similarly, 2 follows the same cycle of 2,4,8,6. Hence, when 44 is divided by 4 it gives a remainder of 0, which corresponds to the last digit being 6 for the second term. 36 would then lead to a difference of 7 and that would be the answer. Hence, option D




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