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What is the digit on the units place in the expanded value of 97^275 –

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

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New post 02 Jul 2019, 21:20
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The units place in the expanded value of 97^275–32^44?
7 raise to any power has cyclicity of 7, 9 , 3, 1 as its unit digit
97^275 = 97^ (272+3) the unit digit for the result would be 3

Similarly, 2 raise to any power has cyclicity of 2, 4 , 8, 6 as its unit digit
32^44 the unit digit for the result would be 6

3-6 --> 13-6--> 7

Answer D
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 02 Jul 2019, 22:01
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We are only concerned about the digit in the units place.
So, even in the above problem \(97^{275}– 32^{44}\) we can only be concerned about \(...7^{275} – ...2^{44}\)

The unit’s digit of powers of 7 are in order of 7, 9, 3, 1
Remainder of \(\frac{275}{4}\) = 3
The unit’s digit of the third power of 7 is 3. Thus, \(7^{275}\) will also have units digit of 3.

Similarly,
The unit’s digit of powers of 2 are in order of 2, 4, 8, 6
Remainder \(\frac{44}{4}\) = 0 i.e 4
The unit’s digit of the fourth power of 2 is 6. Thus \(2^{44}\) will also have the unit’s digit of 6.

Hence,
Unit’s digit of \(7^{275} – 2^{44}\) = Unit’s digit of \(97^{275} – 32^{44}\) = ….3 - ….6 = …7

Answer: D
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 02 Jul 2019, 22:03
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Numbers ending with 7 will end with 7,9,3,1 based on the power. --> Power cycle for 7 is 4.
97^275 will end with 3 since 275 when divided by 4 remainder is 3.

Similarly, any number ending with 2 will have 2/4/8/6 in its unit;s digit based on the power.
32^44 will end with 6, as 44 when divided by 4 remainder is 0.

Also, notice that 97^275 > 32^44. Hence, 97^275 -32^44= .........3 -...........6=............7

OA is D
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 02 Jul 2019, 22:03
A)1

I just tried to get the units digit of 7^5 -2^4
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 02 Jul 2019, 22:04
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To solve this question we will use the pattern method.

We know that we can find the unit digit of any number from 0 to 9 by finding out if the power is of 4k+1, 4k+2, 4k+3 or 4k.

Using this we get that power 275 is of form 4k+3 and 44 is of form 4k.

Unit digit of 7 for power of form 4k+3 is 3 and unit of 2 for power of the form 4k is 6.

The difference between a number of X3 (where X any interger > 0) and 6 will be 7.

Hence, answer is 7.

Answer: D
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 02 Jul 2019, 22:16
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Unit digit of 97^275 would be same as 7^275, which would be same as 7^3 (the unit digits of 7^x repeat after x = 4), which is 3.
Similarly, unit digit of 32^44 would be same as 2^44, which would be same as 2^4 (the unit digits of 2^x repeat after x = 4), which is 6.

Now, ____3-____6 will be same as 13-6 = 7 (where "____x" represents some number string ending with x)
Hence, answer should be (D).
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 02 Jul 2019, 22:29
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Quote:
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


Unit digit of \(97^{275}–32^{44}\) = Unit digit of \(7^{275}–2^{44}\)

CONCEPT: The unit digit of any number depends on the unit digit of the number so any digit other than unit digit is irrelevant such as \(97^{275}\) will have same unit digit as \(387^{275}\) or \(7^{275}\)

Now, Unit digit of \(7^{275}–2^{44}\) = Unit digit of \(7^3–2^4\)

CONCEPT: The cyclicity of unit digit of 7 and 2 is 4 i.e.e unit digits repeat after every 4th power. therefore Remainder (275/4) = 3 hence unit digit of \(7^275\) = unit digit of \(7^3\)

Unit digit of \(7^3–2^4\) = 3-6 = 7

Answer: Option D
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 02 Jul 2019, 22:31
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Unit digits for the powers of 97: 7, 9, 3, 1, 7, 9, 3........ and so on. Cyclicity = 4

Unit digits for the powers of 32: 2, 4, 8, 6, 2, 4, 8........ and so on. Cyclicity = 4

Now, for the 275th power of 97, unit digit will be "3". (275=68*4+3. So, 3rd term in the series [7,9,3,1])

For the 44th power of 32, unit digit will be "6". (44=11*4, 4th term in the series [2,4,8,6])

So, difference for any number with these unit digits will be "7". (X...X3 - X..X6 = X..X7)

ANSWER : D
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 02 Jul 2019, 22:52
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Last digit of the first no is 7 which a cyclicity of 4 . so last digit will be 3.
Similarly last digit of second no is 4. So last digit will be 6.
So effective last digit will be 3-6 =7
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 02 Jul 2019, 23:16
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1
Check for the cyclicity of 7- 3M which means unit digit is going to be 3 and then cyclicity of 2 is 4M which means that the unit digit is going to be 6.

So 3 - 6 which means 13 - 6 = 7 Answer is D
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 02 Jul 2019, 23:21
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97^1, 97^5, 97^9, etc. have a units digit of 7
97^2, 97^6, 97^10, etc. have a units digit of 9
97^3, 97^7, 97^11, etc. have a units digit of 3
97^4, 97^8, 97^12, etc. have a units digit of 1

From the above-mentioned pattern, one can deduce that 97^275 has a units digit of 3.


32^1, 32^5, 32^9, etc. have a units digit of 2
32^2, 32^6, 32^10,etc. have a units digit of 4
32^3, 32^7, 32^11,etc. have a units digit of 8
32^4, 32^8, 32^12,etc. have a units digit of 6

Similarly, one can deduce that 32^44 has a units digit of 6.


Therefore, the digit on the units place 97^275–32^44 is: ...3 - ...6 = ...7

Answer is (D)

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

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New post 03 Jul 2019, 00:52
7 follows a cycle of 4 and 2 follows a cycle of 4
The unit digit is of the two number is (3-2) =1
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 03 Jul 2019, 01:23
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97^275:Since the cyclicity is of 4 numbers(7,9,3,1), we can divide the power, 275, by 4 to get a remainder of 3. so the last digit will be 3
32^44:Since the cyclicity is of 4 numbers(2,4,8,6), we can divide the power, 44, by 4 to get a remainder of 0. so the last digit will be 6.
so the answer should be 7
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 03 Jul 2019, 01:25
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What is the digit on the units place in the expanded value of 97275–3244
97
275

32
44
?

A. 1
B. 3
C. 5
D. 7
E. 9

Since we are looking for the unit's place, let's figure out the unit's places of the numbers involved
97^275 will have 3 at the unit's place (formula: cyclicity of 4)

32^44 will have 6 at the unit's place (formula: cyclicity of 4)

Difference: _ _ _ 3 - _ _ _ 6 = _ _ _ 7

IMO, answer is 7
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 03 Jul 2019, 01:29
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97275–3244

The last digit of the number 97^275 is defined by the the power of 7. The power of 7 has the cycle of 4 numbers (7, 9, 3, 1). => 275 = 4*68 + 3 => the last digit of the number 97^275 is 3.
The last digit of the number 32^44 is defined by the the power of 2. The power of 2 has the cycle of 4 numbers (2, 4, 8, 6). => 44 = 4*11 => the last digit of the number 32^44 is 6.

XXX3 - XXX6 = XXX7 = > Answer is D.
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 03 Jul 2019, 01:37
\(7^5\) = digit in the unit place would be 7(for square- 9, for cube- 3 and so on and so forth)
\(2^4\) = digit in the unit place would be 6 (for square- 4, for cube- 8 and so on and so forth)
therefore, 7-6(subtraction, since the last digit can be subtracted directly) would yield answer 1.
Option A
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 03 Jul 2019, 02:27
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the first part is having units digit as 3, as 7 is having a cyclicity of 4,the power 275 when divided by 4, remainder is 3, so 7^3, last digit is 3;

similarly the second part is having units digit as 6, as cyclicity of 2 is 5, & 44 divided by 5 gives remainder as 4, 2^4, last digit will be 6;

subtraction of XXXX..X3-YYY.YY6; last digit will be 7
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 03 Jul 2019, 02:51
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Using the cyclical nature of unit digits raised to a power, we can easily calculate the answer.
For units digit of 97^275,
7 is the units digit of 97.
We know 7^1=7
7^2=9 (the units digit)
7^3=3
7^4=1
The units digits then repeat in 7,9,3,1 order. the power of 97, 275 is in the form of 4n+3.
So 97^275 would have 3 as units digit.

For units digit of 32^44,
2 is the units digit of 32.
We know 2^1=2
2^2=4
2^3=8
2^4=6(the units digit)
The units digits then repeat in 2,4,8,6 order. the power of 32, 44 is in the form of 4n.
So 32^44 would have 6 as units digit.

So the units digit of the substraction of these numbers would be = 3-6
Since 6 is bigger here than 3, 1 would carry over from the tens side to 3 making it 13.
hence, 13-6=7 is the answer
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 03 Jul 2019, 02:56
It's 7^275-2^44

275/4 will.give a remainder of 3 , so units digit will 3

44/4 will.give a remainder of 0 , unit digit will be 2

3-2=1

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

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New post 03 Jul 2019, 03:25
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imo 7,

question can be written as 7^3 - 2^4 so 3-6 gives ans 7
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Re: What is the digit on the units place in the expanded value of 97^275 –   [#permalink] 03 Jul 2019, 03:25

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