A Square and a Cube: questions with answers

The factors are the powers of 5, from power 0 to power 2. This gives 3 distinct positive factors, including the repeated middle factor only once.

Choose from 0 to 4 copies of 2, and from 0 to 2 copies of 5. There are 5 × 3 = 15 positive factors.

$8 \times 8 = 64$ and $(-8) \times (-8) = 64$. Every positive perfect square has exactly two integer square roots, one positive and one negative.

A square root multiplied by itself gives the original number. Since $(-13) \times (-13) = 169$, $-13$ is a square root of $169$.

17 × 17 × 17 = 4913, which ends in 3. Cubes can end in every digit from 0 to 9.

The first two negative factors give 25. Multiplying by -5 gives -125. Three negative factors have a negative product.

Area is side multiplied by side: 8 × 8 = 64 square centimetres. Adding four sides gives perimeter, not area.

The area is 6 × 6 = 36 square centimetres. Doubling the side or adding its four sides does not measure the surface.

The difference between 1225 and 1296 is 71. Excluding both endpoints leaves one fewer whole number: 70.

10 + 15 = 25 = 5 × 5. These consecutive triangular numbers add to a perfect square. Doubling the smaller gives less than 25; doubling the larger gives more.

For a perfect cube, the prime factors must form triplets of equal factors. Each prime occurs three times, so the factors form complete triplets and the number is a perfect cube. Pairs of equal prime factors test for a perfect square.

4913 lies between 1000 and 8000. Its root is in the 10s. Only a units digit of 7 has a cube ending in 3, giving 17.

Squaring multiplies the number by itself. The nonzero part becomes 7 squared, and 2 trailing zeros become 4, giving 490000.

Split all prime factors into two identical groups. Each group has product 20, and 20 × 20 = 400. Taking each distinct prime only once misses repeated pairs.

24 squared is 576, and 25 squared is 625. Since 619 lies strictly between these squares, its positive square root lies between 24 and 25.