No square number ends in 7
Tonight's idea
A square number is a whole number times itself: 1, 4, 9, 16, 25, 36 and so on. Look at the last digits. You never see 2, 3, 7 or 8. Here is why.
Take 37, which is \(30 + 7\). A square with side 37 splits into four pieces.
Three of the pieces contain a 30, so they end in 0. Only \(7 \times 7 = 49\) can touch the last digit. So \(37^2\) ends in 9, the same as \(7^2\). (It is 1369.) The last digit of any square depends only on the last digit of the number you squared.
That leaves just ten cases to check:
| number ends in | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| its square ends in | 0 | 1 | 4 | 9 | 6 | 5 | 6 | 9 | 4 | 1 |
The bottom row holds only 0, 1, 4, 5, 6 and 9. Every whole number ends in one of the ten digits on top, so nothing has been missed. That is a proof: no square number ends in 2, 3, 7 or 8.
Watch one: spot the fake
Is 4,356,787 a square number? It ends in 7, so no. No calculator needed.
Is 2,209 a square number? It ends in 9, so it might be. The test can only rule numbers out. If it is a square, the table says the number that was squared (its square root) ends in 3 or 7. Now \(40^2 = 1600\) and \(50^2 = 2500\), so try 43 and 47.
\[43^2 = 1849 \qquad 47^2 = 2209\]Yes: 2,209 is \(47^2\).
Your turn
- Which of these cannot be square numbers? Say how you know. 1,443 · 5,776 · 9,218 · 7,225 · 3,002 · 6,889
- 1,936 is a square number. (a) Which two digits could its square root end in? (b) Which two multiples of ten does the root lie between? (c) Find the root.
- Square 5, 15, 25 and 35. What are the last two digits every time? Split 35 into \(30 + 5\) and use the four pieces to explain why.
- A cube is a number times itself times itself, like \(2 \times 2 \times 2 = 8\). (a) Make the ten-case table for cubes. (b) Can you spot a fake cube from its last digit? (c) 2,197 is a cube. What digit does the number that was cubed end in? Find it.
Stretch: the digit next door
The odd squares begin 1, 9, 25, 49, 81, 121, 169, 225. Write the first two as 01 and 09. The tens digit is even every time. Use the four pieces to show it always will be. Then decide, without a calculator: is 3,571,931 a square number?
Answers
- 1,443, 9,218 and 3,002 cannot: they end in 3, 8 and 2. The others pass the test, and in fact \(5776 = 76^2\), \(7225 = 85^2\) and \(6889 = 83^2\).
- (a) 4 or 6. (b) Between 40 and 50, since \(40^2 = 1600\) and \(50^2 = 2500\). (c) \(44^2 = 1936\). (The other candidate gives \(46^2 = 2116\).)
- 25, 225, 625, 1225: always 25. The pieces of \(35^2\) are \(30 \times 30 = 900\), then \(30 \times 5\) twice, which is \(30 \times 10 = 300\), then \(5 \times 5 = 25\). The first three together end in 00 whatever the tens digit is, so only the 25 shows.
- (a) Numbers ending in 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 have cubes ending in 0, 1, 8, 7, 4, 5, 6, 3, 2, 9. (b) No. Every digit appears in the bottom row, so the last digit rules nothing out. (c) Only a number ending in 3 has a cube ending in 7. \(10^3 = 1000\) and \(20^3 = 8000\), so it is 13: \(13^3 = 2197\).
Stretch. Split an odd number into tens and a last digit, say \(30 + 7\). The big piece ends in 00. The two middle pieces are equal, so together they are an even number of tens. The corner piece is 01, 09, 25, 49 or 81, with an even tens digit. Even plus even is even, so the tens digit of an odd square is even. 3,571,931 is odd with tens digit 3, so it is not a square.