I am dividing this method into four parts and we will discuss each part one by one:
a. Last two digits of numbers which end in one
b. Last two digits of numbers which end in 3, 7 and 9
c. Last two digits of numbers which end in 2
d. Last two digits of numbers which end in 4, 6 and 8
Before we start, let me mention Binomial theorem in brief as we need it for calculation
(x+a)^n = Nc0*a^n+Nc1*a^n-1*x+Nc2*a^n-2*x^2+….. where NcR =n!/(r!(n-r)!)
a. Last two digits of numbers which end in one
This is best explained with an example.
What are the last two digits of 31^786?
Solution: 31^786 = (30+1)^786 = 786C0*1^786+786C1*1^785*30+786C2*1^784*30^2…..
Note that in the above expression, after the first two terms, all the terms will have two or more 0’s at the right hand side (since the expression is multiplied by 30^x). Hence to find the last two digits, only first two terms needs to be evaluated. First term equals to 1 and second term equals to 786*30 = 23580 hence last two digits will be 81 (which is last two digits of 23580 and +1 hence 80+1)
Wiyee!! That’s really long isn’t it? Well those who need a shortcut of this read on!!
Multiply the tenth digit ( 3 in the above example) with the unit digit of the exponent (6 in the above example) to get the tenth digit. Unit digit is always one when a number ends in 1.
Now it’s a two second job isn’t it??
Few exercises for you:
41^2789
71^56747
51^456*61^567
Post your answers using comment option
Now I have covered 25% of this exercise i.e. part A alone. Based on your response, if you find this useful, I will go ahead and post the approach for below kind of numbers
b. Last two digits of numbers which end in 3, 7 and 9
c. Last two digits of numbers which end in 2
d. Last two digits of numbers which end in 4, 6 and 8
Thanks,
Quant-Master
Okay!! So you now know how to find the unit digit of a number of the form x^y that ends in 1, 2, 5, 6,.
Before proceeding just see if you can get answers to the below problems
1) 128975^478z
2) 1928321^67846y
3) 8726352^1234
4) 876192736^xyz
Wiyee!! You got all correct than let’s proceed. If you are not sure about any of the above answer than have a look at my post on finding unit digit of a number – Part 1
Now let’s have a look at the number 4 and 9.
Why I selected these two particular numbers?
Just like 1, 5 and 6 these two numbers also have something in common.
Both 4 and 9 have cycles of 2.
4^1 = 4
4^2 =16
4^3 =64
4^4 = 256
Ahh! There you see. Unit digit repeats itself for every two powers. So 4^2k+1 will have 4 as unit digit and 4^2k will have unit digit as 6. To put it in more simple words, 4 raised to even power will have 6 as unit digit and 4 raised to odd power will have 4 as unit digit.
Now if you calculate for 9 it will also yield similar pattern
9^1 = 9
9^2 = 81
9^3 =729
9^4 = 6561
So like you can see 9^x will have unit digit 9 or 1. 9 raised to odd power will have 9 as unit digit and 9 raised to even power will have 1 as unit digit.
So we are done with numbers 1,2,4,5,6 and 9.
Let’s have a look at 3,7 and 8.
What do these numbers have in common?
Just like 2, they have a cycle of 4.
Let’s try 3
3^1 = 3
3^2 = 9
3^3 =27
3^4 = 81
3^5 = 243
Okay! If you can see the unit digit repeating after 3^4, you got the trick.
3^4k+1 will end in 3
3^4k+2 will end in 9
3^4k+3 will end in 7
3^4k will end in 1
Similarly for 7 and 8
7^4k+1 will end in 7
7^4k+2 will end in 9
7^4k+3 will end in 3
7^4k will end in 1
8^4k+1 will end in 8
8^4k+2 will end in 4
8^4k+3 will end in 2
8^4k will end in 6.
There ends my responsibility. Now it’s your turn.
Find out the unit digit for following numbers
1625251^xyz
124352^5416
12735373^267255
1636284^2a
16273835^n
1763826^2n
172637^27635
1738273268^172282
17283639^2b+1
Let me know if you have any questions. My next post will be on permutation and combination.
Happy blogging,
Quant-Master1)If x and y are integers and x*y^2 is a positive odd integer, which of the following must be true?
Ⅰ. xy is positive.
Ⅱ. xy is odd.
Ⅲ. x + y is even.
(A) Ⅰ only
(B) Ⅱ only
(C) Ⅲ only
(D) Ⅰ and Ⅱ
(E) Ⅱ and Ⅲ
2)If a committee of 3 people is to be selected from among 5 married couples so that the committee does not include two people who are married to each other, how many such committees are possible?
A. 20
B. 40
C. 50
D. 80
E. 120
3)Last Sunday a certain store sold copies of Newspaper A for $1.00 each and copies of Newspaper B for $1.25 each, and the store sold no other newspapers that day. If rpercent of the store’s revenues from newspaper sales was from Newspaper A and if ppercent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p?
A. 100p / (125 – p)
B. 150p / (250 – p)
C. 300p / (375 – p)
D. 400p / (500 – p)
E. 500p / (625 – p)
4)Alice's take home pay last year was the same each month and she saved the same fraction of her take home pay each month.The total amount of money that she had saved at the end of the year was 3 times the amount of that portion of her monthly take home pay that she did not save. If all the money that she saved last year was from her take home pay, what fraction of her take home pay did save each month.
A 1/2
B 1/3
C 1/4
D 1/5
E 1/6
5)A certain university will select 1 of 7 candidates eligible to fill a position in the mathematics department and 2 of 10 candidates eligible to fill 2 identical positions in the computer science department. If none of the candidates is eligible for a position in both departments, how many different sets of 3 candidates are there to fill the 3 positions?
A. 42
B. 70
C. 140
D. 165
E. 315
6)On Saturday morning, Malachi will begin a camping vacation and he will return home at the end of the first day on which it rains. If on the first three days of the vacation, the probability of rain on each day is 0.2, what is the probability that Malachi will return home at the end of the day on the following Monday?
(A) 0.008
(B) 0.128
(C) 0.488
(D) 0.512
(E) 0.640
7)The perimeter of a certain isosceles right triangle is 16+16 sqr root 2.
What is the lenght of the hypotenuse of the triangle.
A 8
B 16
C 4 sqr root 2
D 8 sqr root 2
E 16 sqr root 2
8)The sum of three integers is 40. The largest integer is 3 times the middle integer, and the smallest integer is 23 less than the largest integer. What is the product of the three integers?
A. 1,104
B. 972
C. 672
D. 294
E. 192
9)If n is a multiple of 5 and n=p^2q, where p and q are prime numbers, which of the following must be a multiple of 25?
(a) p^2
(b) q^2
(c) pq
(d) p^2q^2
(e) p^3q
10)The number 75 can be written as the sum of the squares of 3 different positive intergers.What is the sum of these 3 integers.
a 17
b 16
c 15
d 14
e 13
Official answers will be posted by friday. Post/Discuss the answers using comment option. Here is a comparison chart to gauge your performance
| Correct | Remarks |
| 9+ | Good |
| 7-8 | Above Average |
| 4-6 | Average |
| <4 | Need to work harder |
Posts
All Comments
