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What is the unit digit of 7^6 -6^7 22 Apr 2019, 20:28
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B
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Difficulty:

95% (hard)

Question Stats:

26% (01:28) correct 74% (00:51) wrong based on 203 sessions

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What is the unit digit of \(7^6 - 6^7\)?

A. 3
B. 7
C. 5
D. 1
E. 9
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B
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What is the unit digit of 7^6 -6^7 22 Apr 2019, 21:02
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What is the unit digit of \(7^6 - 6^7\)?

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

The term \(7^6=7^(4+2)\), so same units digit as 7^2 or units digit is 9.
The term \(6^7\) will end in 6 ..

So, \(7^6 - 6^7\) is same as( ..abc9 - ..xyz6)
So two cases..
(1) \(7^6 >6^7\), then units digit is 9-6 or 3
(2) \(7^6 <6^7\), then units digit is 16-9 or 7

Now \(a^b>b^a\), when a and b are consecutive and b>a, that is b=a+1

So here case II

B
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What is the unit digit of 7^6 -6^7 22 Apr 2019, 21:32
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chetan2u
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What is the unit digit of \(7^6 - 6^7\)?

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

The term \(7^6=7^(4+2)\), so same units digit as 7^2 or units digit is 9.
The term \(6^7\) will end in 6 ..

So, \(7^6 - 6^7\) is same as( ..abc9 - ..xyz6)
So two cases..
(1) \(7^6 >6^7\), then units digit is 9-6 or 3
(2) \(7^6 <6^7\), then units digit is 16-9 or 7

Now \(a^b>b^a\), when a and b are consecutive and b>a, that is b=a+1

So here case II

B
Show more


Are there any exceptions to \(a^{b}>b^a\), when a and b are consecutive and b>a, that is b=a+1?

I tried on numbers 2 & 3, 3= 2+1 , but \(2^3(8) < 3^2(9)\)
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What is the unit digit of 7^6 -6^7 22 Apr 2019, 21:40
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manu11
chetan2u
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What is the unit digit of \(7^6 - 6^7\)?

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

The term \(7^6=7^(4+2)\), so same units digit as 7^2 or units digit is 9.
The term \(6^7\) will end in 6 ..

So, \(7^6 - 6^7\) is same as( ..abc9 - ..xyz6)
So two cases..
(1) \(7^6 >6^7\), then units digit is 9-6 or 3
(2) \(7^6 <6^7\), then units digit is 16-9 or 7

Now \(a^b>b^a\), when a and b are consecutive and b>a, that is b=a+1

So here case II

B


Are there any exceptions to \(a^{b}>b^a\), when a and b are consecutive and b>a, that is b=a+1?

I tried on numbers 2 & 3, 3= 2+1 , but \(2^3(8) < 3^2(9)\)
Show more

Hi..
Yes the initial two positive integers are exceptions..
1^2<2^1
2^3<3^2
But thereafter the gaps in two keeps increasing the other way..

3^4>4^3...81>64
4^5>5^4..1024>625
And so on..
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What is the unit digit of 7^6 -6^7 22 Apr 2019, 22:52
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Excellent observation buddy.
There is one more way to check which term is greater
assume \(7^6\)>\(6^7\)
or \((7/6)^6\)>6
or \((1.1667)^6\)>6
Now we know \((1.1667)^6\)<\((1.2)^6\)
\((1.1667)\)^6<\((1.44)^3\)<\((1.5)^3\)
\((1.1667)^6\)<3.375
So our assumption was wrong
\(7^6\) must be smaller than \(6^7\)


chetan2u
nick1816
What is the unit digit of \(7^6 - 6^7\)?

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

The term \(7^6=7^(4+2)\), so same units digit as 7^2 or units digit is 9.
The term \(6^7\) will end in 6 ..

So, \(7^6 - 6^7\) is same as( ..abc9 - ..xyz6)
So two cases..
(1) \(7^6 >6^7\), then units digit is 9-6 or 3
(2) \(7^6 <6^7\), then units digit is 16-9 or 7

Now \(a^b>b^a\), when a and b are consecutive and b>a, that is b=a+1

So here case II

B
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What is the unit digit of 7^6 -6^7 04 Feb 2023, 15:40
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Can someone please explain to me in details why is 6^7 greater than 7^6? Is it some rule related to consecutive integers that I have just to learn by heart? It is hard for me to memorise rules that I don't understand. Thank you in advance.
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What is the unit digit of 7^6 -6^7 05 Feb 2023, 00:55
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HoudaSR
Can someone please explain to me in details why is 6^7 greater than 7^6? Is it some rule related to consecutive integers that I have just to learn by heart? It is hard for me to memorise rules that I don't understand. Thank you in advance.
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It's not a formula as such, but obviously it's a TRAP people fall into! Keep a bookmark of this problem and just remember that in subtractions, it's important to check which term is greater before we go into any sort of calculations!
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What is the unit digit of 7^6 -6^7 28 Mar 2025, 00:43
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In question,they ask for the unit digit of the subtraction between 7^6 and 6^7.So why the given answer is B when the question doesn't ask for 7^6<6^7?
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What is the unit digit of 7^6 -6^7 28 Mar 2025, 02:04
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AyshaAkter001
In question,they ask for the unit digit of the subtraction between 7^6 and 6^7.So why the given answer is B when the question doesn't ask for 7^6<6^7?
Show more
There could be 2 possible answers for units digit based on which of the either \(7^6\) or \(6^7\) is greater.

If \(7^6\) is greater than \(6^7\) then \(7^6 - 6^7\) = units digit is \(3\); example \(7^2 - 6\) = \(49 - 6\) = 43

and,

If \(6^7\) is greater than \(7^6\) then \(7^6 - 6^7\) = units digit is \(7\); example \(7^2 - 6^3\) = \(49 - 216\) = -167

Therefore, it is necessary to find out which one is greater; and since \(6^7 > 7^6\), the answer is B. 7
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