Exercise practice 1: A Square and a Cube
A Square and a Cube24 min readLesson 3 of 6
3.1 Exercise practice 1: Figure it Out, page 10
Every question from the first Figure it Out section is worked below. Read the statement, try it yourself, then check the steps.
3.1.1 E1.1 Not perfect squares
Question. From the list 2032, 2048, 1027 and 1089, identify every number that cannot be a perfect square.
Answer. 2032, 2048 and 1027 are not perfect squares. 1089 is a perfect square because 1089 = 33^2.
Key steps. A perfect square never ends in 2, 3, 7 or 8. The units digits here are 2, 8, 7 and 9. The first three numbers end in forbidden digits, so they are ruled out at once. For 1089, estimate: it lies between 32^2 = 1024 and 34^2 = 1156, so try 33^2 = 1089, which matches. Checking the units digit alone cannot prove a number is a square; the multiplication does.
3.1.2 E1.2 Last digit 4
Question. Which of 64^2, 108^2, 292^2 and 36^2 has last digit 4?
Answer. 108^2 and 292^2.
Key steps. Work with the units digit of the base. A number ending in 2 or 8 has a square ending in 4. The bases 108 and 292 end in 8 and 2, so their squares end in 4. The other two bases end in 4 and 6, and their squares end in 6. Check: 108^2 = 11664 and 292^2 = 85264; 64^2 = 4096 and 36^2 = 1296.
3.1.3 E1.3 Using 125 squared
Question. Given 125^2 = 15625, which choice gives 126^2? The options add 126, 262, 253, 251 or 512 to 15625.
Answer. The fourth choice, 15625 + 251, because 126^2 = 15876.
Key steps. The difference between consecutive squares is the sum of the two numbers: 126^2 - 125^2 = 125 + 126 = 251. Equivalently, the next odd number after the first 125 odd numbers is 2 x 126 - 1 = 251. So add 251 to 15625 to get 15876.
3.1.4 E1.4 Side of a square
Question. A square covers 441 square metres; what is its side length?
Answer. 21 m.
Key steps. The side is the square root of the area. Since 20^2 = 400 and 22^2 = 484, the root lies between 20 and 22. Test 21^2 = 441, which equals the area. So the side is 21 m.
3.1.5 E1.5 Smallest square divisible by 4, 9 and 10
Question. Find the smallest square number divisible by 4, 9 and 10.
Answer. 900.
Key steps. First find the least common multiple: LCM(4, 9, 10) = 180 = 2^2 x 3^2 x 5. For a number to be a square, every prime exponent must be even. Here the exponent of 5 is 1, so multiply by 5: 180 x 5 = 900 = 2^2 x 3^2 x 5^2 = (2 x 3 x 5)^2 = 30^2. So 900 is the smallest square divisible by all three numbers.
3.1.6 E1.6 Making 9408 a square
Question. What is the least multiplier that turns 9408 into a perfect square, and what is the square root of the result?
Answer. Multiply by 3; the product is 28224 = 168^2.
Key steps. Factorise: 9408 = 2^6 x 3 x 7^2. The exponents 6 and 2 are even, but the 3 has exponent 1. Multiplying by 3 makes the exponent 2. The product is 2^6 x 3^2 x 7^2 = (2^3 x 3 x 7)^2 = 168^2. Check: 168 x 168 = 28224.
3.1.7 E1.7 Counting between squares
Question. Count the whole numbers strictly between 16^2 and 17^2, and then between 99^2 and 100^2.
Answer. 32 numbers and 198 numbers.
Key steps. Between n^2 and (n+1)^2 the count is 2n. For 16 and 17, the count is 2 x 16 = 32. For 99 and 100, the count is 2 x 99 = 198. Equivalently, subtract and then remove one endpoint: 17^2 - 16^2 - 1 = 289 - 256 - 1 = 32, and 100^2 - 99^2 - 1 = 10000 - 9801 - 1 = 198.
3.1.8 E1.8 Missing numbers in a pattern
Question. Fill the blanks: 4^2 + 5^2 + 20^2 = ()^2 and 9^2 + 10^2 + ()^2 = (__)^2.
Answer. 4^2 + 5^2 + 20^2 = 21^2, and 9^2 + 10^2 + 90^2 = 91^2.
Key steps. In each row the first two bases are consecutive numbers n and n + 1; the third base is their product n(n + 1); the right side is n(n + 1) + 1 squared. Check row one: 4 x 5 = 20 and 4 x 5 + 1 = 21, and indeed 16 + 25 + 400 = 441 = 21^2. For row two, 9 x 10 = 90 and 90 + 1 = 91; then 81 + 100 + 8100 = 8281 = 91^2.
3.1.9 E1.9 Counting tiny squares
Question. How many tiny squares are in the picture, and what is the prime factorisation of that number?
Answer. 2025 tiny squares; 2025 = 3^4 x 5^2.
Key steps. The picture is a 9 by 9 array of tiles, so there are 81 tiles. Each tile, whether it is drawn as a square or as a diamond, is a 5 by 5 grid of tiny squares, so each tile contains 25 tiny squares. Total = 81 x 25 = 2025. Factorising: 2025 = 81 x 25 = 3^4 x 5^2. Notice that 2025 = 45^2, a perfect square.