Power Mad
In this investigation, I will explore the way powers of numbers behave.
1. Work out 21, 22, 23,24, 25, 26…
Power of 2 Answer Units Digit
21 2 2
22 4 4
23 8 8
24 16 6
25 32 2
26 64 4
27 128 8
28 256 6
2. For which values of n will 2n be a multiple of 10?
As can be seen from the following table, the unit digits of the powers of two are in a repetitious pattern of 2, 4, 8, 6, 2, 4, 8, 6…
All multiples of 10 have a unit digit of 0. However, as seen from the pattern, none of the powers of 2 have their unit digit ending in a 0. Therefore, no power of 2 is a multiple of 10.
3. Work out 2n+3n for different odd values of n.
What do you notice?
n 2n+3n
1 5
3 35
5 275
7 2315
9 20195
11 179195
13 1602515
I noticed that the value of 2n+3n, where n is an odd number, always ends in a 5. I can deduce that this can only happen if both the unit digit of 2n and 3n results in 5. To demonstrate this, I worked out the unit digits of both 2n and 3n.
The powers of 2’s unit digit go in a pattern of 2, 4, 8, 6. The power of 3’s unit digit goes in a pattern of 3, 9, 7, 1. When n is an odd number, the unit digit of the powers of 2 is either a 2 or an 8, as the unit digits go up in a pattern of 2, 4, 7, 6. When n is an odd number, the unit digit of the powers of 3 is either 3 or 7, as the unit digits go up in a pattern of 3, 9, 7, 1. Therefore, the unit digit of the value of 2n+3n when n is an odd number is either 2+3, or 8+7, which results in 5.
4. What do you notice when n is a multiple of 4?
When n is a multiple of 4:
n 2n+3n
4 97
8 6817
12 535537
16 4112257
20 3487832977
I noticed that the value of 2n+3n, where n is multiple of 4, always ends in a 7. I can deduce that this can only happen if both the unit digit of 2n and 3n results in 7. To demonstrate this, I worked out the unit digits of both 2n and 3n. This is because:
The powers of 2’s unit digits go in a pattern of: 2, 4, 8, 6.
The powers of 3’s unit digits go in a pattern of: 3, 9, 7, 1.
When n is a multiple of 4, the unit digit of the powers of 2 is always a 6, as the unit digits of the powers of 2 go up in a pattern of 2, 4, 8, 6. When n is a multiple of 4, the unit digit of the powers of 3 is always a 1, as the unit digits of the powers of 3 go up in a pattern of 3, 9, 7, 1. Therefore, the unit digit of 2n+3n is always 1 + 6, which equals to a 7.
For what values is 1n + 2n + 3n even?
Next I will find for which values of n is 1n + 2n + 3n even:
n 1n + 2n + 3n
1 6
2 14
3 36
4 98
5 276
I noticed that the values of 1n + 2n + 3n is always even no matter what the value of n is. This is because:
1n is always a 1 (an odd number). The value of 2n is always an even number (the unit digits of 2n is always 2, 4, 8, or 6, which means that 2n is always an even number).
The value of 3n is always an odd number (the unit digits of 3n is always 3, 9, 7, or 1, which means that 3n is always an odd number). When an odd number (1n), an even number (2n), and an odd number (3n) are added together, the result is always an even number. This can be demonstrated by saying that 2x+1 is an odd number, and 2x is an even number.
(2x+1) + (2x+1) + (2x) = 6x + 2 = 2(3x + 1), which equals an even number, as 2 times any number is even.
6. Work out 1n + 2n + 3n + 4n for different values of n. What do you notice?
n 1n + 2n + 3n + 4n
1 10
2 30
3 100
4 354
5 1300
6 4890
7 18700
8 72354
9 282340
10 1108650
11 4373500
12 17312754
I noticed that all values of 1n + 2n + 3n + 4n end in a 0. However, when n is a multiple of 4, the result ends in a 4 instead of a 0.
This is because:
The unit digit of 1n is always 1.
The unit digit of 2n is in a pattern of 2, 4, 8, 6…
The unit digit of 3n is in a pattern of 3, 9, 7, 1…
The unit digit of 4n is in a pattern of 4, 6, 4, 6…
When n is 1, the sum of the unit digits of the four numbers is:
1 + 2 + 3 + 4 = 10
When n is 2, the sum of the unit digits of the four numbers is:
1 + 4 + 9 + 6 = 20
When n is 3, the sum of the unit digits of the four numbers is:
1 + 8 + 7 + 4 = 20
When n is 4, the sum of the unit digits of the four numbers is:
1 + 6 + 1 + 6 = 14
Therefore, the values of 1n + 2n + 3n + 4n always end in 0, with the exception of when n is a multiple of 4, for which then the unit digit of 1n + 2n + 3n + 4n is 4.
7. What about 1n + 2n + 3n + 4n + 5n?
n 1n + 2n + 3n + 4n + 5n
1 15
2 55
3 225
4 979
5 4425
6 20515
7 46825
8 462979
I noticed that all values of 1n + 2n + 3n + 4n + 5n end in a 5. However, when n is a multiple of 4, the result ends in a 9 instead of a 5.
This is because:
The unit digit of 1n is always 1.
The unit digit of 2n is in a pattern of 2, 4, 8, 6…
The unit digit of 3n is in a pattern of 3, 9, 7, 1…
The unit digit of 4n is in a pattern of 4, 6, 4, 6…
The unit digit of 5n is in a pattern of 5, 5, 5, 5…
When n is 1, the sum of the unit digits of the five numbers is:
1 + 2 + 3 + 4 + 5 = 15
When n is 2, the sum of the unit digits of the five numbers is:
1 + 4 + 9 + 6 + 5 = 25
When n is 3, the sum of the unit digits of the five numbers is:
1 + 8 + 7 + 4 + 5 = 25
When n is 4, the sum of the unit digits of the five numbers is:
1 + 6 + 1 + 6 + 5 = 19
Therefore, the values of 1n + 2n + 3n + 4n + 5n always end in 5, with the exception of when n is a multiple of 4, for which then the unit digit of 1n + 2n + 3n + 4n is 9.
8. What other surprising results can you find?
The unit digit of 1n (or a base number with the unit digit of 1) is in a pattern of 1, 1, 1, 1…
The unit digit of 2n (or a base number with the unit digit of 2) is in a pattern of 2, 4, 8, 6…
The unit digit of 3n (or a base number with the unit digit of 3) is in a pattern of 3, 9, 7, 1…
The unit digit of 4n (or a base number with the unit digit of 4) is in a pattern of 4, 6, 4, 6…
The unit digit of 5n (or a base number with the unit digit of 5) is in a pattern of 5, 5, 5, 5…
The unit digit of 6n (or a base number with the unit digit of 6) is in a pattern of 6, 6, 6, 6…
The unit digit of 7n (or a base number with the unit digit of 7) is in a pattern of 7, 9, 3, 1…
The unit digit of 8n (or a base number with the unit digit of 8) is in a pattern of 8, 4, 2, 6…
The unit digit of 9n (or a base number with the unit digit of 9) is in a pattern of 9, 1, 9, 1…
The unit digit of 10n (or a base number with the unit digit of 0) is in a pattern of 0, 0, 0, 0…
Thus, whatever number, the unit digits of the powers are all in a pattern in groups of four, so all powers have their unit digit in a repetitious pattern in groups of 4, as all numbers end with the above 10 numbers. Let a number be x. The pattern of the unit digits of the powers of x is a, b, c, d... Let another number be y. The pattern of the unit digits of the powers of y is e, f, g, h…
As a result, the sum/difference of the unit digits of the powers of x and y are in a repetitious pattern of: a + e, b + f, c + g, and d + h, or a - e, b - f, c - g and d – h.
As a result, no matter what combination of powers, adding or subtracting, the unit digit’s results would always be in a pattern of 4 repetitious numbers.