Worked examples
The question
Question: The pattern red, blue, green, red, blue, green, ...
keeps repeating.
What is the 50th color?
1 Find the block
The list starts again after green.
So the block that repeats is red, blue, green.
block = red, blue, green
The block is 3 long.
length = 3
2 Divide
The block comes again and again.
So we divide by its length.
What is left over tells us the place in the block.
n ÷ (length of the block): what is left = the place in the block
3 times 16 is 48.
So 16 full blocks take us to the 48th color.
50 minus 48 is 2 more.
50 ÷ 3 = 3 × 16 = 48, 50 − 48 = 2
3 Count into the block
Careful!
The 16 only counts full blocks.
The 2 left over gives the place.
Careful: not 16: the number of blocks ✗
2 left means place 2 in the block: blue.
50th color = place 2 in the block = blue
4 Check
Let's check by counting on.
A block ends at the 48th color.
Then we count on: red, blue.
The 50th is blue.
So it is right!
48th = green (end of a block), 49th = red, 50th = blue ✓
Let's look back.
First: find the block.
Then: divide.
Then: count into the block.
Last: check.
Answer 50th color = place 2 in the block = blue
▶ Watch it solved
The question
Question: what is the ones digit of 7⁴2?
ones digit of 742 = ?
1 Find the block
7 to the power 1 ends in 7.
71 = ends in 7
The ones digit of the next power comes only from the ones digit times 7.
So we keep just the ones digits.
only the ones digit matters: ones digit × 7
7 to the power 2: 7 times 7 is 49.
It ends in 9.
72 = 7 × 7 = 49: ends in 9
7 to the power 3: 9 times 7 is 63.
It ends in 3.
73 = 9 × 7 = 63: ends in 3
7 to the power 4: 3 times 7 is 21.
It ends in 1.
74 = 3 × 7 = 21: ends in 1
7 to the power 5: 1 times 7 is 7.
It ends in 7 again: we are back at the start.
75 = 1 × 7 = 7: ends in 7
So the ones digits repeat: 7, 9, 3, 1, again and again.
block = 7, 9, 3, 1
The block is 4 long.
length = 4
2 Divide
The block comes again and again.
So we divide by its length.
What is left over tells us the place in the block.
n ÷ (length of the block): what is left = the place in the block
4 times 10 is 40.
So 10 full blocks take us to the 40th power.
42 minus 40 is 2 more.
42 ÷ 4 = 4 × 10 = 40, 42 − 40 = 2
3 Count into the block
Careful!
The 10 only counts full blocks.
The 2 left over gives the place.
Careful: not 10: the number of blocks ✗
2 left means place 2 in the block: 9.
ones digit of 742 = place 2 in the block = 9
4 Check
Let's check by counting on.
A block ends at the 40th power.
Then we count on: 7, 9.
The 42nd is 9.
So it is right!
40th = 1 (end of a block), 41st = 7, 42nd = 9 ✓
Let's look back.
First: find the block.
Then: divide.
Then: count into the block.
Last: check.
Answer ones digit of 742 = place 2 in the block = 9
▶ Watch it solved
The question
Question: What is the 50th digit after the decimal point of 1/7?
1 Find the block
10 divided by 7 is 1, and 3 is left.
digit 1 = 10 ÷ 7 = 1, 3 left
We divide 1 by 7 the long way.
Each time we bring down a 0.
1 ÷ 7: bring down a 0 each time
30 divided by 7 is 4, and 2 is left.
digit 2 = 30 ÷ 7 = 4, 2 left
20 divided by 7 is 2, and 6 is left.
digit 3 = 20 ÷ 7 = 2, 6 left
60 divided by 7 is 8, and 4 is left.
digit 4 = 60 ÷ 7 = 8, 4 left
40 divided by 7 is 5, and 5 is left.
digit 5 = 40 ÷ 7 = 5, 5 left
50 divided by 7 is 7, and 1 is left.
1 left: that is where we started.
So the digits repeat from here.
digit 6 = 50 ÷ 7 = 7, 1 left
So 1/7 is 0 point 1 4 2 8 5 7, and the same 6 digits again and again.
17 = 0.142857 142857 …
The block is 6 long.
length = 6
2 Divide
The block comes again and again.
So we divide by its length.
What is left over tells us the place in the block.
n ÷ (length of the block): what is left = the place in the block
6 times 8 is 48.
So 8 full blocks take us to the 48th digit.
50 minus 48 is 2 more.
50 ÷ 6 = 6 × 8 = 48, 50 − 48 = 2
3 Count into the block
Careful!
The 8 only counts full blocks.
The 2 left over gives the place.
Careful: not 8: the number of blocks ✗
2 left means place 2 in the block: 4.
50th digit = place 2 in the block = 4
4 Check
Let's check by counting on.
A block ends at the 48th digit.
Then we count on: 1, 4.
The 50th is 4.
So it is right!
48th = 7 (end of a block), 49th = 1, 50th = 4 ✓
Let's look back.
First: find the block.
Then: divide.
Then: count into the block.
Last: check.
Answer 50th digit = place 2 in the block = 4
▶ Watch it solved
