Divisor: Difference between revisions
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imported>Richard L. Peterson m (bolded and tookout caps on "Does Not", tenses etc) |
imported>Richard L. Peterson ("We stress that" instead of "Note that", indenting, etc) |
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Given two [[integer]]s ''d'' and ''a'', where ''d'' is nonzero, d is said to ''divide a'', or ''d'' is said to be a ''divisor'' of ''a'', if and only if there is an integer ''k'' such that ''dk = a''. For example, 3 divides 6 because 3*2 = 6. Here 3 and 6 play the roles of ''d'' and ''a'', while 2 plays the role of ''k''. | Given two [[integer]]s ''d'' and ''a'', where ''d'' is nonzero, d is said to ''divide a'', or ''d'' is said to be a ''divisor'' of ''a'', if and only if there is an integer ''k'' such that ''dk = a''. For example, 3 divides 6 because 3*2 = 6. Here 3 and 6 play the roles of ''d'' and ''a'', while 2 plays the role of ''k''. | ||
More examples:6 is a divisor of 24 since 6*4 = 24. ( | More examples: | ||
:6 is a divisor of 24 since 6*4 = 24. (We stress that ''6 divides 24'' and ''6 is a divisor of 24'' mean the same thing.) | |||
5 divides 0 because 5*0 = 0. In fact, every integer except zero divides zero. | :5 divides 0 because 5*0 = 0. In fact, every integer except zero divides zero. | ||
7 is a divisor of 49 since 7*7 = 49. | :7 is a divisor of 49 since 7*7 = 49. | ||
7 divides 7 since 7*1 = 7. | :7 divides 7 since 7*1 = 7. | ||
1 divides 5 because 1*5 = 5. In fact, 1 and -1 divide every integer. | :1 divides 5 because 1*5 = 5. In fact, 1 and -1 divide every integer. | ||
2 '''does not''' divide 9 because there is no integer k such that 2*k = 9. Since 2 is not a divisor of 9, 9 is said to be an [[odd]] integer, or simply an [[odd]] number. | :2 '''does not''' divide 9 because there is no integer k such that 2*k = 9. Since 2 is not a divisor of 9, 9 is said to be an [[odd]] integer, or simply an [[odd]] number. | ||
*Note that 0 is never a divisor of any number. For example, if 0 were to divide 8, there would have to be an integer ''k'' such that ''0*k = 8'', which is impossible. (Nor does 0 divide 0, by convention rather than impossibility.) | |||
Revision as of 00:58, 30 March 2007
Divisor (Number theory)
Given two integers d and a, where d is nonzero, d is said to divide a, or d is said to be a divisor of a, if and only if there is an integer k such that dk = a. For example, 3 divides 6 because 3*2 = 6. Here 3 and 6 play the roles of d and a, while 2 plays the role of k.
More examples:
- 6 is a divisor of 24 since 6*4 = 24. (We stress that 6 divides 24 and 6 is a divisor of 24 mean the same thing.)
- 5 divides 0 because 5*0 = 0. In fact, every integer except zero divides zero.
- 7 is a divisor of 49 since 7*7 = 49.
- 7 divides 7 since 7*1 = 7.
- 1 divides 5 because 1*5 = 5. In fact, 1 and -1 divide every integer.
- 2 does not divide 9 because there is no integer k such that 2*k = 9. Since 2 is not a divisor of 9, 9 is said to be an odd integer, or simply an odd number.
- Note that 0 is never a divisor of any number. For example, if 0 were to divide 8, there would have to be an integer k such that 0*k = 8, which is impossible. (Nor does 0 divide 0, by convention rather than impossibility.)