Find the unit digit of (432)^{412} × (499)^{431}
Given:
(432)^{412} × (499)^{431}
Concept:
9^{even no.} = unit digit 1
9^{odd no.} = unit digit 9
Calculation:
(432)^{412} × (499)^{431}
Taking unit digits
⇒ 2^{412} × 9^{431}
As we know unit digit of
2^{1} = 2, 2^{2} = 4, 2^{3} = 8, 2^{4} = 6
⇒ 2^{4(103)} × 9^{431}
⇒ 6 × 9
⇒ 54
∴ The unit digit of (432)^{412} × (499)^{431} is 4.
What is the units digit of (a +b)^{2} – (ab)^{2}, where a and b are nonnegative integers?
(1) The difference between any two consecutive multiples of a is 5
(2) b when multiplied by any even integer results in the same units digit, not necessarily equal to units digit of b.
Steps 1 & 2: Understand Question and Draw Inferences
Step 3: Analyze Statement 1 independently
If a = 5, 4ab = 20b. So, irrespective of the value of b, the units digit(20b) = 0.
Sufficient to answer.
Step 4: Analyze Statement 2 independently
2. b when multiplied by any even integer results in the same units digit, not necessarily equal to units digit of b.
As we have a unique answer of Units Digit(4ab), this statement is sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
What is the units digit of z^{x}, where x and z are positive integers?
(1) z when divided by 100 has its hundredths digit as 5
(2) The product of z^{2} and z^{3} has the same units digit as z^{2}.
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Units Digit (z^{x})
Step 3: Analyze Statement 1 independently
1. z when divided by 100 has its hundredths digit as 5
The hundredths digit of z100
Now, a number with its Units Digit = 5, raised to any power will always have its Units Digit as 5.
So, Units Digit (z^{x}) = 5
Sufficient to answer.
(Note: Even if you took z to have, say, 8 digits – a number like z = abcdefgh, where a to h are digits – you would still get the units digit of z (the units digit is h in this case) as 5. Therefore, the answer would still be the same)
Step 4: Analyze Statement 2 independently
2. The product of z^{2} and z^{3} has the same units digit as z^{2}.
As we do not have a unique value of Units Digit of(z), this statement is insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step3, this step is not required.
Answer: A
For any positive number n, the function #n represents the value of the number n rounded to the nearest integer. If k is a positive number, what is the units digit of #k?
(1) #(10k) = 10k
(2) #(100k) is 10300.
Steps 1 & 2: Understand Question and Draw Inferences
Given:
To find:
Thus, to answer the question, we need to know the value of a and whether b < 5 or not
Step 3: Analyze Statement 1 independently
(1) #(10k) = 10k
10k = 10n + 10a + b.cde. . .
10k = 10(n+a) + b.cde. . .
So, Statement 1 alone is not sufficient.
Step 4: Analyze Statement 2 independently
(2) #(100k) is 10300.
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 4, this step is not required
Answer: Option B
Arrange the following terms in the increasing order of their units digits:
I. 7^{5}
II. 8^{6}
III. 12^{3}
Given: 3 terms: 7^{5}, 8^{6} and 12^{3}
To find: The correct increasing order of the units digits of the 3 terms
Approach:
Working Out:
For any positive number x, the function [x] denotes the greatest integer less than or equal to x. For example, [1] = 1, [1.367] = 1 and [1.999] = 1.
If k is a positive integer such that k^{2} is divisible by 45 and 80, what is the units digit of
Given:
To find: The units digit of
Approach:
Working Out:
Looking at the expression for k^{3} above, we see that k^{3} will be divisible by 4000
Looking at the answer choices, we see that the correct answer is Option E
n is a positive integer that lies between 100 and 200, exclusive, and has no digits repeated. Is the units digit of n equal to 5?
(1) n is divisible by 9 and the 2digit number formed by inverting the units and tens digit of n is a prime number.
(2) The difference between the units digit and the tens digit of n is the same as the difference between the tens digit and the hundreds digit of n and is equal to 2.
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Is Units Digit(n) = 5?
Step 3: Analyze Statement 1 independently
Hence, the possible values of n = { 135, 198}. So, the units digit of n = {5, 8}
As we do not have a unique value of units digit(n), the statement is insufficient to answer.
Step 4: Analyze Statement 2 independently
2. The difference between the units digit and the tens digit of n is the same as the difference between the tens digit and the hundreds digit of n and is equal to 2.
Thus, the units digit(n) = 5
As we have a unique value of units digit(n) , this statement is sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique value from step 4, this step is not required.
Answer: B
If p and q are positive integers, what is the remainder when (27)^{12p} + (3)^{6q} is divided by 5?
(1) p is an odd prime number
(2) q is divisible by 10
Steps 1 & 2: Understand Question and Draw Inferences
Given: Positive integers p and q
To find:
So, to answer the question, we need to know if q is even or odd.
Step 3: Analyze Statement 1 independently
(1) p is an odd prime number
Statement 1 gives no information about q’s evenodd nature.
So it’s not sufficient to answer the question
Step 4: Analyze Statement 2 independently
(2) q is divisible by 10?
Since we now know the evenodd nature of q, we can answer the question. Statement 2 alone is sufficient.
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 4, this step is not required
Answer: Option B
If x and y are distinct positive integers and x+y is even, what is the remainder when (x+y)^{a} is divided by 10, where a is a positive integer?
(1) Units digit of y is 6
(2) (xy)^{a} is divisible by 10.
Steps 1 & 2: Understand Question and Draw Inferences
To Find: the value of r in (x+y)^{a}=10k+r, where k is the quotient obtained when (x+y)^{a} is divided by 10 and r is the remainder; so, 0 ≤ r < 10
Step 3: Analyze Statement 1 independently
1. Units digit of y is 6
Hence, insufficient to answer.
Step 4: Analyze Statement 2 independently
2. (xy)^{a} is divisible by 10.
Insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
Combining both the statements, we can say that units digit (x) = 0 and units digit(y) = 6
So, units digit of (x+y) = 6. Now do we need the value of a to find out the units digit of (x+y)?
We know a number with units digit of 6 raised to any power always results in units digit of 6.
So, Units Digit ((6^{a})) = 6.
Thus r = Units Digit ((6^{a})) = 6.
Sufficient to answer.
Answer: C
If a and b are positive integers, is the sum a + b divisible by 4?
Steps 1 & 2: Understand Question and Draw Inferences
Given: Integers a, b > 0
To find: If a + b is divisible by 4?
Step 3: Analyze Statement 1 independently
Statement 1 says that ‘When the sum 23^{a} + 25^{b} is divided by 10, the remainder is 8’
Step 4: Analyze Statement 2 independently
Statement 2 says that ‘When 22^{b} is divided by 10, the remainder is 8’
Step 5: Analyze Both Statements Together (if needed)
Answer: Option C
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