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If p is a positive integer, what is the units digit of Z, if Z = (1044p + 1) * (277p + 1) * (93p + 2) * (3096p) ?
- a)2
- b)6
- c)7
- d)8
- e)9
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
If p is a positive integer, what is the units digit of Z, if Z = (1044...
Step 1: Question statement and Inferences
We are given that Z = (1044p + 1) * (277p + 1) * (93p + 2) * (3096p). We have to find the units digit of Z.
Here we can say that:
The unit digit of Z = The units digit of the product of the unit digits of the given numbers
Now, we also know that the unit digit of any power of a number depends only on the unit digit of the number. Thus, we can write the expression as:
Z = (44p + 1) * (7p + 1) * (3p + 2) * (96p)
Step 2: Finding required values
Z = (44p + 1) * (7p + 1) * (3p + 2) * (96p)
Next, let’s find the unit digit of the individual expressions:
Unit digit of 44p + 1:
Every second power of 4 has the same unit digit.
Cycles of powers of 4 are 4, 6, 4, 6 …
So, unit digit of 44p + 1 = 4 (Since 4p + 1 is an odd number and every odd power of 4 has the unit digit as 4)
Unit digit of 96p:
Every second power of 9 has the same unit digit.
Cycles of powers of 9 are 9, 1, 9, 1 …
So, unit digit of 96p = 1 (Since 6p is an even number and every even power of 9 has the unit digit 1)
Now, the cyclicity of the numbers 3 and 7 is 4. So, we can’t decide the unit digit of the expression 3p+2 and 7p+1 since we don’t know the value of p. However, the product of these numbers can be further solved as follows:
7p + 1 * 3p + 2 = 7p + 1 * 3p + 1 * 3
= (7*3)p + 1 * (3) (Since am * bm = (ab)m )
= (21)p + 1 * (3)
Now, we know that the unit digit of the expression 21p+1 will always be 1 since any power of 1 always gives a unit digit 1.
Thus, the unit digit of (21)p + 1 * (3) = 1 * 3 = 3
Step 3: Calculating the final answer
Now, let’s plug in all the values in the expression for Z.
Z = 4 * 3 * 1 = 12
So, the unit digit of Z will be 2.
Answer: Option (A)
Most Upvoted Answer
If p is a positive integer, what is the units digit of Z, if Z = (1044...
To find the units digit of Z, we need to find the units digit of each term in the expression and then multiply them together.
Given expression: Z = (1044p - 1) * (277p - 1) * (93p - 2) * (3096p)
Finding the units digit of each term:
1. (1044p - 1):
- The units digit of 1044p will always be 4, as the units digit of any number to the power of a positive integer is the same as the units digit of the base number.
- Subtracting 1 from 4 gives us 3 as the units digit.
2. (277p - 1):
- The units digit of 277p will always be 7, as explained above.
- Subtracting 1 from 7 gives us 6 as the units digit.
3. (93p - 2):
- The units digit of 93p will always be 3, as explained above.
- Subtracting 2 from 3 gives us 1 as the units digit.
4. (3096p):
- The units digit of 3096p will always be 6, as explained above.
Multiplying the units digits together:
3 * 6 * 1 * 6 = 108
Since we are only interested in the units digit, we can ignore the tens digit (0) and the hundreds digit (1), and focus only on the units digit, which is 8.
Therefore, the units digit of Z is 8, and the correct answer is option A.
Given expression: Z = (1044p - 1) * (277p - 1) * (93p - 2) * (3096p)
Finding the units digit of each term:
1. (1044p - 1):
- The units digit of 1044p will always be 4, as the units digit of any number to the power of a positive integer is the same as the units digit of the base number.
- Subtracting 1 from 4 gives us 3 as the units digit.
2. (277p - 1):
- The units digit of 277p will always be 7, as explained above.
- Subtracting 1 from 7 gives us 6 as the units digit.
3. (93p - 2):
- The units digit of 93p will always be 3, as explained above.
- Subtracting 2 from 3 gives us 1 as the units digit.
4. (3096p):
- The units digit of 3096p will always be 6, as explained above.
Multiplying the units digits together:
3 * 6 * 1 * 6 = 108
Since we are only interested in the units digit, we can ignore the tens digit (0) and the hundreds digit (1), and focus only on the units digit, which is 8.
Therefore, the units digit of Z is 8, and the correct answer is option A.
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