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14, 49, −21 and 0 are multiples of 7, whereas 3 and −6 are not. This is because there are integers that 7 may be multiplied by to reach the values of 14, 49, 0 and −21, while there are no such integers for 3 and −6.
Animation demonstrating the smallest Pythagorean triple, 3 2 + 4 2 = 5 2. A Pythagorean triple consists of three positive integers a, b, and c, such that a 2 + b 2 = c 2. Such a triple is commonly written (a, b, c), a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k.
Cycles of the unit digit of multiples of integers ending in 1, 3, 7 and 9 (upper row), and 2, 4, 6 and 8 (lower row) on a telephone keypad. Figure 1 is used for multiples of 1, 3, 7, and 9. Figure 2 is used for the multiples of 2, 4, 6, and 8. These patterns can be used to memorize the multiples of any number from 0 to 10, except 5.
Four bags with three marbles per bag gives twelve marbles (4 × 3 = 12). Multiplication can also be thought of as scaling. Here, 2 is being multiplied by 3 using scaling, giving 6 as a result. Animation for the multiplication 2 × 3 = 6 4 × 5 = 20. The large rectangle is made up of 20 squares, each 1 unit by 1 unit.
The sequence (a k) of least numbers n such that σ(n) > kn, in which a 2 = 12 corresponds to the first abundant number, grows very quickly (sequence A134716 in the OEIS). The smallest odd integer with abundancy index exceeding 3 is 1018976683725 = 3 3 × 5 2 × 7 2 × 11 × 13 × 17 × 19 × 23 × 29. [8]
The starting play is determined either by who holds the heaviest (highest) double (or single, if no one has a double) and that person plays first. If it is a 6–4, 5–5, 5–0, 4–1, or 3–2, the initial count is evenly divisible by five and so the player scores. Players in turn then lay a matching tile on one of the endpoints.
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Note that numbers that will be discarded by a step are still used while marking the multiples in that step, e.g., for the multiples of 3 it is 3 × 3 = 9, 3 × 5 = 15, 3 × 7 = 21, 3 × 9 = 27, ..., 3 × 15 = 45, ..., so care must be taken dealing with this.