S-BATCH @ SRICHAITANYA: unitarydigits

Explaination

Explanation for Question1:

What could be the units digit of 2²ⁿ, where n is an integer.

Answer – (C) Solution: 2²ⁿ is nothing but 2^even
2^even ends with either 4 or 6.

Find the units digit of 4^{3^4²}

Answer – (C) Solution:
4^{3^4²} is nothing but 4^odd.
4^odd always ends with 4.

What is the unit digit of 1¹x2²X3³ X.......9⁹

Answer – (A) Solution: 1¹x2²X3³ X.......9⁹ consists 5⁵
5⁵ ends with 5.
Multiples of 5 either end with 0 or 5.
5 multipled with even numeber always end with zero.
So the answer is 0.

Explanation for Question 4:

What could be the units digit of 1947³ⁿ

Solution:
the units digit of 1947³ⁿ is same as the units digit of 7³ⁿ
when n=1, the units digit of 7³=3
when n=2, the units digit of 7⁶=9
when n=3, the units digit of 7⁹=7
when n=4, the units digit of 7¹²=1

5.What is the unit digit of 9³ⁿ⁺⁵, n is a prime number greater than 5..
Answer:6
Solution: Any prime number greater than 5 is always odd.
therefore 3n is odd
3n+5 is even.
9³ⁿ⁺⁵=9^even,which always ends with 1.

Explanation for Question 6:
.Find the units digit of 1001¹+1002²+1003³+.......1009⁹?

Answer – The units digit of1¹=1
The units digit of2²=4
The units digit of3³=7
The units digit of4⁴=6
The units digit of5⁵=5
The units digit of6⁶=6
The units digit of7⁷=3
The units digit of8⁸=6
The units digit of9⁹=9
Sum of the units digit =47

Explanation for Question 7:
What could be the units digit of 9ⁿX 2²ⁿ, where n is an integer?
Answer :By plugging n=1,2,3...in 9ⁿX 2²ⁿ,it always ends with 6.

Explanation for Question8:
Find the units digit of 1¹X3³X5⁵X7⁷X.......97⁹⁷X99⁹⁹
Answer:5⁵ ends with 5.
Multiples of 5 either end with 0 or 5.
5 multipled with odd numeber always end with 5.
So the answer is 5.

Explanation for Question9:
abcd is a four digit number. Find the value of d if the units digit of (abcd)¹⁵ is 6?
Answer-by verifying the options d= 6 satisfies. so the answer is 6.

Explanation for Question 10: abcd is a four digit number. What could be the value of d if the units digit of (abcd)¹² is 6
Solution:By verifying the optios d=2,4,6,8 satisfies.

Explanation for Question 11:
172²⁷¹ has a units digit of:

Answer – 8
Solution: the units digit of 172²⁷¹ is equal to 2²⁷¹
2²⁷¹ is in the form of 2⁴ⁿ⁺³,which end with 8.So the answer is 8.

Explanation for Question 12:
a,b are single digit positive integers. If the units digit of a^b+b^a=6,then find the value of (a+b).
. Solution: The only combination satisfies is 5,1. Hence the answer is 6.

Explanation for Question 13:1¹+2²+3³+...+100¹⁰⁰ is divided by 2. What is the remainder?
Explanatory Answer
1¹+2²+3³+...+100¹⁰⁰ is even.
When even number is divided by 2 it leaves a remainder 0.

Explanation for Question 14:What is the units digit of (71)⁵(46)³(103)⁴ + (57)(1088)³
Explanatory Answer
The units digit of 71⁵=7
The units digit of 46³=6
The units digit of 103⁴=1
The units digit of(71)⁵(46)³(103)⁴ = 1 x 6 x 1=6---------> (i)
The units digit of 1088³=2
The units digit of (57)(1088)³= 7 x 2 =4---------->(ii)
adding (i) and (ii) the answer is 0.

Explanation for Question 15
What is the units digit of 177²⁸ – 133²³?
Explanatory Answer
the units digit of 177²⁸=1
the units digit of 133²³=7
1-7=4 ( 177²⁸ is greater than 133²³)

Explanation for Question 16: If a,b are two positive integers which of the following must be true.
Solution:A:
A. 5^a+6^b is even :5^a is odd,6^b is even
odd+even=odd, option A is ruled out.
B. 8^a+6^b is odd:8^a is even,6^b is even
even+even=even, option B is ruled out.
C. 5^a+7^b is odd:5^a is odd,7^b is odd
odd+odd=even, option C is ruled out.
D. 7^a+6^b is divisible by2.:7^a is odd,6^b is even
odd+even=odd, which is not divisible by 2.
E. 5^a+6^b is odd. 5^a is odd,6^b is even
odd+even=odd, option E is the correct.

Explanation for Question 17:
How many zeroes does 200²⁰ have?
Explanatory Answer 200²⁰=2²⁰x100²⁰
100²⁰ ends with 40 zeroes.

Explanation for Question 18: What is the first non-zero digit that the number 200²⁰has from the right?
Explaination: 200²⁰=2²⁰x100²⁰
2²⁰ ends with 6.
the first non-zero digit in 200²⁰ is 6.

Explanation for Question 19: What is the units digit of 29²⁰⁰⁸-18¹⁹⁹⁶
Explaination:the units digit of 29^{2008=1
the units digit of18¹⁹⁹⁶ is 6.
29²⁰⁰⁸ is greater than 18¹⁹⁹⁶
11-6=5(1 carring 1 from the prevous digit 1 will become 11)}

Explanation for Question 20: What is the units digit of 29²⁰⁰⁸-18¹⁹⁹⁶
Explaination:the units digit of 29^{2008=1
the units digit of18¹⁹⁹⁶ is 6.
29²⁰⁰⁸ is greater than 18¹⁹⁹⁶
1+6=7.}

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