hey.. can anyone pls tell me what is the answer and solving method for the question:
unit's digit of 7^(5^(7^5))
thanks..
hey.. can anyone pls tell me what is the answer and solving method for the question:
unit's digit of 7^(5^(7^5))
thanks..
My take 7...
hey.. can anyone pls tell me what is the answer and solving method for the question:
unit's digit of 7^(5^(7^5))
thanks..
The answer should be 7
hey.. can anyone pls tell me what is the answer and solving method for the question:
unit's digit of 7^(5^(7^5))
thanks..
It will be 7.
If we look at powers of 7 i.e. 7^1, 7^2, 7^3 etc, we find that the last digits follow the pattern 7, 9, 3, 1, 7, 9, 3, 1...repeating after every 4 steps - i.e. the cyclicity of 7 is 4 steps.
So we need to take the given power (in this case (5^(7^5))) and see what happens when this is divided by 4. Now if we observe carefully, this is of the form 5^x which is nothing but (4+1)^x, which on binomial expansion will give all terms divisible by 4 except the last term 1. In other words we can say that (5^(7^5)) is of the form 4k+1.
Hence 7^(4k+1) can be written as (7^4k) * (7^1) = (7^4)^k * (7^1) using laws of indices. But (7^4) ends in 1 and hence (7^4)^k will also end in 1, while (7^1)ends in 7 so that when they are multiplied the last digit will be 1 * 7 = 7.
Now comes the fun part, learning how to do all this orally!
~scrabbler
thanks.. dude
Progressions SET-6 :-
1) If a+b, b+c and c+a are in harmonic progression, then
a) a^2, b^2 and c^2 will be in AP
b) a^2, bc and c^2 will be in HP
c) 2ab, a^2 and b^2 will be in AP
d) b^2, a^2 and c^2 will be in AP
2) If 1/(1+x) , 1/(1-x), 4/(1-x^2) are in Arithmetic progression, then
a) x lies between 0 and 2
b) x lies between -2 and 0
c) x lies between 2 and 4
d) there is no possible values of x
3) If the sides of a right-angled triangle are in AP, then the minimum possible difference between the magnitude of the area and perimeter of such a triangle is..
a) 0
b) 6
c) 12
d) 4.5
4) What is the sum of the product of all possible pairs of distinct numbers of the first ten natural numbers.
a) 1320
b) 2640
c) 1705
d) 1920
5) If log 0.5 , log(2^x -1) and log(2^x+5) are in arithmetic progression, then x=
a) 3/2
b) log 3 base 2
c) log 2 base 3
d) - log 2 base 3
It will be 7.
If we look at powers of 7 i.e. 7^1, 7^2, 7^3 etc, we find that the last digits follow the pattern 7, 9, 3, 1, 7, 9, 3, 1...repeating after every 4 steps - i.e. the cyclicity of 7 is 4 steps.
So we need to take the given power (in this case (5^(7^5))) and see what happens when this is divided by 4. Now if we observe carefully, this is of the form 5^x which is nothing but (4+1)^x, which on binomial expansion will give all terms divisible by 4 except the last term 1. In other words we can say that (5^(7^5)) is of the form 4k+1.
Hence 7^(4k+1) can be written as (7^4k) * (7^1) = (7^4)^k * (7^1) using laws of indices. But (7^4) ends in 1 and hence (7^4)^k will also end in 1, while (7^1)ends in 7 so that when they are multiplied the last digit will be 1 * 7 = 7.
Now comes the fun part, learning how to do all this orally!
~scrabbler
i think it can also be done like this :
7^5^7^5=>
taking unit digit of powers from up to down=> consider 7^5 its unit digit will be 7 as cyclicity of 7 is 4.....
now unit digit of 5^(the number whose unit digit is 7) is 5...as we multiply 5 as many times we get unit digit 5
now unit digit of 7^(the number whose unit digit is 5)=> cyclicity of 7 is 4...and any number ending with unit digit 5 divided by 4 will have remainder 1...so unit digit of 7^(5^(7^5)) will be 7
4) What is the sum of the product of all possible pairs of distinct numbers of the first ten natural numbers.
a) 1320
b) 2640
c) 1705
d) 1920
1705 my take
4) What is the sum of the product of all possible pairs of distinct numbers of the first ten natural numbers.
a) 1320
b) 2640
c) 1705
d) 1920
1705 my take
No it is not correct
i think it can also be done like this :
7^5^7^5=>
taking unit digit of powers from up to down=> consider 7^5 its unit digit will be 7 as cyclicity of 7 is 4.....
now unit digit of 5^(the number whose unit digit is 7) is 5...as we multiply 5 as many times we get unit digit 5
now unit digit of 7^(the number whose unit digit is 5)=> cyclicity of 7 is 4...and any number ending with unit digit 5 divided by 4 will have remainder 1...so unit digit of 7^(5^(7^5)) will be 7
That will not work....the last assumption you made is itself wrong "any number ending with unit digit 5 divided by 4 will have remainder 1" - for example consider 15 divided by 4, leaves remainder 3! Units digit alone is not sufficient - this is an easy trap to fall into (speaking from experience here 😃 )
~scrabbler
4) What is the sum of the product of all possible pairs of distinct numbers of the first ten natural numbers.
a) 1320
b) 2640
c) 1705
d) 1920
1705 my take
Should be 1320
~scrabbler
profit and loss :
1) i purchased a dining table for rs 8000 two years ago. present price of a similar dining table is rs 15000. if i sell my table, i would get rs 10000 for it. if instead i polish it, i would have to spend rs 3000 on it but can sell it at the price of a new one. at what discount should i sell my dining table now if i expect the same percentage profit as i would have in the event of my selling it without polishing ?
a - 25/3 %
b - 10 %
c - 23/2 %
d - 77/5 %
2) i purchased items worth rs 1500. i sold some items and found out that i had made a profit of 20% which incidentally was equal to 5% of my total purchase. what is the woth of the items that i had sold ?
a- 225
b- 375
c- 675
d- 1125
please explain the approach too. thanks
1) i purchased a dining table for rs 8000 two years ago. present price of a similar dining table is rs 15000. if i sell my table, i would get rs 10000 for it. if instead i polish it, i would have to spend rs 3000 on it but can sell it at the price of a new one. at what discount should i sell my dining table now if i expect the same percentage profit as i would have in the event of my selling it without polishing ?
a - 25/3 %
b - 10 %
c - 23/2 %
d - 77/5 %
2) i purchased items worth rs 1500. i sold some items and found out that i had made a profit of 20% which incidentally was equal to 5% of my total purchase. what is the woth of the items that i had sold ?
a- 225
b- 375
c- 675
d- 1125
please explain the approach too. thanks
help pls..
Q1. Which numbers have exactly 3 factors?
Q2. x, a and b are natural numbers. If x is divisible by a and beach then find a condition for which x is definitely divisible by a*b?
Q3.If LCM of a, b, c is x and that of d, e is y then, would the LCM of x,y be the LCM of a, b, c, d, e.
Similarly for HCF what is the answer?
thanks..
help pls..
Q1. Which numbers have exactly 3 factors?
thanks..
all perfect squares of primes have exactly 3 factors
remember all natural numbers have even number of factors except for perfect squares which have odd number of factors.
profit and loss :
1) i purchased a dining table for rs 8000 two years ago. present price of a similar dining table is rs 15000. if i sell my table, i would get rs 10000 for it. if instead i polish it, i would have to spend rs 3000 on it but can sell it at the price of a new one. at what discount should i sell my dining table now if i expect the same percentage profit as i would have in the event of my selling it without polishing ?
a - 25/3 %
b - 10 %
c - 23/2 %
d - 77/5 %
2) i purchased items worth rs 1500. i sold some items and found out that i had made a profit of 20% which incidentally was equal to 5% of my total purchase. what is the woth of the items that i had sold ?
a- 225
b- 375
c- 675
d- 1125
please explain the approach too. thanks
1.Profit % in 1st case :
10000-8000=2000
2000/8000=25%
.Profit % in 2nd case :
15000-11000=4000
4000/11000=?
to make profit %=25
11000+25*11000=13750,But S.P=15000.so he has to give a discount of 1250 on 15000 to make the sp =13750
so disount%=1250/15000=8.33%
2. Let the CP of his purchase be worth x
1/5*x=75 Given
x=375
Can any one post me the approach for this type of question:
How to make 5 people sit in 10 chairs so that no 2 people sit adjacent.Here n+r-1Cr-1 doesnt work.How to go about this? Please illustrate the approach.
Can any one post me the approach for this type of question:
How to make 5 people sit in 10 chairs so that no 2 people sit adjacent.Here n+r-1Cr-1 doesnt work.How to go about this? Please illustrate the approach.
Hi Freakonomist :)
n+r-1 C r-1 won't work because that formula is used to distribute n identical things in r groups.
To solve your question, number the chairs 1 through 10.
If the chairs are to be occupied in such a way that only alternate ones are occupied, there are two possibilities that the 5 people can be distributed in the 10 chairs:
1,3,5,7,9 are occupied OR
2,4,6,8,10 are occupied.
There is no other way of seating the people in different chairs such that no 2 people sit together.
In either case, no. of ways of seating 5 people = 5! = 120
So, total no. of ways = 240 😃
Hi guys,
Can anyone tell me about the weightage of quant and DI in paper one i.e how many questions out of 30 are of quant and how many out of 30 are of DI? Also, tell the distribution for paper II i.e how many questions out of 30 are of VA and how many out of 30 are of LR?
Let me know the approximate cut-off for both the sections as well.
Thanks - Sameer
Can any one post me the approach for this type of question:
How to make 5 people sit in 10 chairs so that no 2 people sit adjacent.Here n+r-1Cr-1 doesnt work.How to go about this? Please illustrate the approach.
_x_x_x_x_x_
out of 6 dash we need to select 5....so 6C5*5!
Hi Freakonomist
n+r-1 C r-1 won't work because that formula is used to distribute n identical things in r groups.
Thanks for the reply,what would have been your approach if it was 3 people in 10 chairs with no adjacent sitting..