(2) 38432718 In the multiplier we first encounter a 0, so we place a 0 in the first line under it, and proceed to multiply by the 3. We again come to a 0, we therefore place another 0 in the 2nd line under it. We then multiply by the 8 and afterwards by the 7. In multiplying by 7 we place the first figure (8) in the 3rd line under the 7. (1) 97643127 37, The sign X, connecting two quantities, signifies that they are to be multiplied together. Ex. 21. (1) 49678 × 2701, 856 (3) 869721 × 5715, 2031 (5) 430618432×5076, 8162 (7) 9876053172038, 7164 (9) 81654321 × 8972, 5436 (11) 89476583×9276, 5430 (13) 91534652×807061, 20765 (15) 912357612378, 43078 (17) 4389216X9176, 94607 (33) 123456789×123456,7891011 (47) 9184327×810526, 819674 38. The multiplier and multiplicand are called factors 39. To resolve a given product into factors, is to find (1) 28, 36, 16, 72, 55, 121 | (18) 394, 210, 70, 35, 32, 49 Ex. 24. Work the following examples by resolving the multiplier into factors. (1) 487106 x 12, 15, 18, 21, 24, 27, 33, 36 (9) 3173891 x 16, 20, 24, 28, 32, 36, 44, 48 (17) 37246393 x 20, 25, 30, 35, 40, 45, 55, 60 (25) 66177188 x 24, 30, 36, 42, 48, 54, 66, 72 (33) 171819311 x 28, 35, 42, 49, 56, 63, 77, 84 (41) 81873472 x 32, 40, 48, 56, 64, 72, 88, 96 (49) 71425541 x 36, 45, 54, 63, 72, 81, 99, 108 (57) 81037690 x 40, 50, 60, 70, 80, 90, 110, 120 (65) 45719321 x 44, 55, 66, 77, 88, 99, 121, 132 (73) 36714736 × 48, 60, 72, 84, 96, 108, 132, 144 42. The square of a number is found by multiplying the number by itself. Find the squares of Ex. 25. (1) 54, 25, 386, 749, 2860|(14) 5707500, 2138607, 1260943 (6) 41387, 168613, 21469, 2867 (17) 214687, 41506312, 384186 (10) 3984, 76038, 71284, 78056 (20) 4138715, 456234, 57002 43. The cube of a number is found by multiplying together the number and its square. Find the cubes of Ex. 26. (1-6) 8, 9, 17, 45, 85, 213] (14-16) 2170860, 2730504, 1860730 (7-10) 854, 6705, 2715, 38679 (17-19) 4150607, 2860384, 168060 (11-13) 21703, 86070, 54083 (20-22) 1730730, 1560738, 215067 Ex. 27. (1) Find the sum of twenty-eight seventeens. (2) In a building there are 43 windows; in each window 15 panes; each pane cost 8d. How many pence did the windows cost? (3) What is a factor? Write two factors of 84. What is a prime factor? Write all the prime factors of 84. (4) Explain what is meant by 8d. x7, and write this down as an addition sum. (5) What will be the cost of making a road 89 miles long, at £28 a yard, there being 1760 yards in a mile? (6) If 17 men do a piece of work in 25 days, how long will it take 1 man to do? (7) Multiply the sum of all the prime numbers up to 40 by the sum of all the prime numbers between 40 and 60. (8) A farmer bought 395 oxen at £29 each, and 187 others at £45 each: would he gain or lose by selling the whole at £35 each ? (9) How many miles will be passed over by a train in 19 hours, if it travel 34 miles an hour? (10) Show that whether you multiply 835 by 120, or by the factors of 120, the result will be the same. (11) Why is it absurd to ask a person to multiply 11 pence by 9 pence? (12) A man's pulse beats 68 times to the minute: how many times will it beat in 75 hours, in each of which there are 60 minutes? (13) The smaller of two numbers is 786, the sum of the two, one thousand nine hundred and eight. Find their product. (14) From the product of eighteen millions seven hundred and fifteen thousand by nineteen thousand four hundred and twenty, take the product of eighteen thousand seven hundred and fifteen by one hundred and ninety-four thousand two hundred. (15) Find the sum of 2795 and 3608, then find their difference, lastly multiply the product of the sum and difference by 19 less than 85. (16) Multiply the sum of 84, 9386, 714985, 2136, and 90135 by the difference between 81354 and 25259. (17) A woman bought 19 hens at 28 pence each, they cost her 4 pence each per week for food. At the end of the first week two died, and at the end of the second week she sold the others at 36 pence each. How many pence did she lose? (18) Find the sum of the six products formed by the six pairs you can make out of the numbers 386, 498,5709, 8607. (19) Multiply the sum of 87968 and 37559 by their difference. Additional Exercises. The following exercises will furnish many thousands of multiplication sums, with verifications of their answers. (1) From the table on page 10, take out any two numbers; square them and subtract the smaller product from the larger. Again multiply the sum of the two numbers by their difference; this result should be the same as the first. (2) Take out of the table any two numbers, multiply them together, and square the products. Again square the numbers, and multiply together the numbers thus obtained. This result should be the same as the first. (3) Take any two numbers and find their sum and difference; square each of these and add the result. Again square each number separately and add the products. The latter sum should be half the former. (4) Take as a multiplicand a line of figures from any of the preceding addition or subtraction sums. Take for multiplier a number from Part I. of the table on page 10; also with the same multiplicand take the corresponding figures from Part II as a multiplier. Add this product to the former, and then add the multiplicand. The answer will be the multiplicand followed by ciphers. (5) Take any number from Part I. of the addition table, and the corresponding number from Part II.; increase the former by 1, and square both numbers. Add the results, and the answer will contain as many ciphers as there were figures in the original numbers, preceded by the difference of these numbers. Simple Division. 44. Simple Division is the process by which we find how often one number is contained in another. 45. The number by which we divide is called the divisor; the number divided is called the dividend: the number of times the divisor is contained in the dividend is called the quotient. 46. Division is the inverse operation to Multiplication. In Multiplication we have a multiplier and a multiplicand to find their product; in Division we have the product of two numbers and one of the numbers to find the other. Thus multiplier multiplicand: product. pro In multiplication either the multiplier or multiplicand must be an abstract number (32), and when one of them is concrete the duct is concrete. From this it follows that if a concrete number be divided by an abstract number, the answer must be concrete; but |