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22:20
Last Digit
Mockat Admin22:20
Module 5 of 8
4. Last Digits
To find the last digits in the product of certain numbers, we can simply multiply the last digits of each of these numbers.In other words, to find the units digit of a product of numbers, we should multiply only their units digits. This is shown in the example below.
In this section we will use the following functions
1) (), which provides the units digit or the last digit of .
2) (), which provides the tens and units digit or the last digits of .
Example 13
What is the units digit of ?
= 6
Answer:
Solution
= 6
Answer:
Similarly, to find the last digits in a product, we find the product of the last digits.
Example 14
What are the last two digits of ?
Answer:
Solution
Answer:
4.1 Units Digit Cyclicity in Powers
To find the last digit for a number expressed as an exponent, say , we need to apply the power only to the last digit of .Example 15
What is the units digit of ?
; ;
;
The units digits are alternating between and . In fact, the units digits are 4 for odd powers and 6 for even powers of .
In , the power of is an odd number. ∴ The units digit is .
Answer:
Solution
To find the units digit of this exponent, we look at powers of the units digit alone. Units digit of is .; ;
;
The units digits are alternating between and . In fact, the units digits are 4 for odd powers and 6 for even powers of .
In , the power of is an odd number. ∴ The units digit is .
Answer:
In the above example, the units digit is repeating for every second power of . This repetition or cyclicity varies across the different digits of our decimal number system.
For instance, raised to the powers of , , , and are , , , and respectively. The last digits of these powers, i.e. \bm{3, 9, 7, 1 \text{and} 3}, are filled where in the table below.
The last digits of consecutive powers of are , , , , , , , , , , ... Note that the last digit for every fourth power is repeated. Therefore, when the last digit is , the cyclicity is .
| Last digit of where | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| Cyclicity | Digits |
|---|---|
The above tables need not be memorised. All you need to note here is that the cyclicity is a maximum of 4 and that the last digits can be quickly found. The same is shown in the example below.
Example 16
What is the units digit of ?
; .
Cyclicity of is . ∴ is 6.
(Starts repeating here)
Cyclicity for is . When the powers of are of the form , , and , where is a natural number, the units digits are , , and respectively.
When is divided by , the remainder is . So, is of the form k + .
∴ is that of , which is 3.
Units digits of , , and are , , and respectively. Cyclicity for is also .
When is divided by , the remainder is . So, is of the form k.
∴ is that of , which is .
(Note: When the remainder is , we have to take units digit of and not .)
Units digits of , , and are , , and respectively. Cyclicity for is .
Units digit is for odd powers and for even powers.
∴ is .
∴
Answer:
Solution
We need to find the units digit of; .
Cyclicity of is . ∴ is 6.
(Starts repeating here)
Cyclicity for is . When the powers of are of the form , , and , where is a natural number, the units digits are , , and respectively.
When is divided by , the remainder is . So, is of the form k + .
∴ is that of , which is 3.
Units digits of , , and are , , and respectively. Cyclicity for is also .
When is divided by , the remainder is . So, is of the form k.
∴ is that of , which is .
(Note: When the remainder is , we have to take units digit of and not .)
Units digits of , , and are , , and respectively. Cyclicity for is .
Units digit is for odd powers and for even powers.
∴ is .
∴
Answer:
Example 17
What is the last digit of ?
We need to find the remainder when is divided by . The following is covered under Factors & Remainders lessson.
∴ is of the form .
Units digit of
Answer:
Solution
The cyclicity for is also . The units digits of , , , are , , and respectively.We need to find the remainder when is divided by . The following is covered under Factors & Remainders lessson.
∴ is of the form .
Units digit of
Answer: