6. If two different numbers be divisible by any one number, their sum and their difference will also be divisible by that number. 7. If several different numbers, divisible by 3, be added or multiplied together, their sum and their product will also be divisible by 3. 8. If two numbers divisible by 9, be added together, their sum of the figures in the amount will be either 9 or a number divisible by 9. 9. If any number be multiplied by 9, or by any other number divisible by 9, the amount of the figures of the product will be either 9 or a number divisible by 9. 10. In every arithmetical progression, if the first and last term be each multiplied by the number of terms, and the sum of the two products be divided by 2, the quotient will be the sum of the series. 11. In every geometric progression, if any two terms be multiplied together, their product will be equal to that term which answers to the sum of these two indices. Thus, in the series: 1 2 3 4 5 2 4 8 16 32 If the third and fourth terms, 8 and 16, be multiplied together, the product, 128, will be the seventh term of the series. In like manner, if the fifth term be multiplied into itself, the product will be the tenth term; and if that sum be multiplied into itself, the product will be the twentieth term. Therefore, to find the last, or twentieth term of a geometric series, it is not necessary to continue the series beyond a few of the first terms. Previous to the numerical recreations, we shall here describe certain mechanical methods of performing arithmetical calculations, such as are not only in themselves entertaining, but will be found more or less useful to the young reader. To Find a Number Thought of. FIRST METHOD. EXAMPLE. Let a person think of a number, say 6 1. Let him multiply by 3 18 2. Add 1 19 3. Multiply by 3 57 4. Add to this the number thought of 63 Let him inform you what is the number produced; it will always end with 3. Strike off the 3, and inform him that he thought of 6. SECOND METHOD. EXAMPLE. Suppose the number thought of to be 6 1. Let him double it 12 2. Add 4 16 3. Multiply by 5 80 4. Add 12 92 5. Multiply by 10 920 Let him inform you what is the number produced. You must then, in every case, subtract 320; the remainder is, in this example, 600; strike off the 2 ciphers, and announce 6 as the number thought of. THIRD METHOD. Desire a person to think of a number--say 6. He must then proceed: