问题11。证明任何正奇数整数的形式为6q +1或6q + 3或6q + 5,其中q是某个整数。
解决方案:
a = bq+r ; where 0 < r < b
Putting b=6 we get,
⇒ a = 6q + r, 0 < r < 6
r = 0, a = 6q = 2(3q) = 2m, which is an even number. [m = 3q]
r = 1, a = 6q + 1 = 2(3q) + 1 = 2m + 1, which is an odd number. [m = 3q]
r = 2, a = 6q + 2 = 2(3q + 1) = 2m, which is an even number. [m = 3q + 1]
r = 3, a = 6q + 3 = 2(3q + 1) + 1 = 2m + 1, which is an odd number. [m = 3q + 1]
r = 4, a = 6q + 4 = 2(3q + 2) + 1 = 2m + 1, which is an even number. [m = 3q + 2]
r = 5, a = 6q + 5 = 2(3q + 2) + 1 = 2m + 1, which is an odd number. [m = 3q + 2]
Therefore, any odd positive integer can be of the form 6q +1,6q + 3,6q + 5, where q is some integer.
问题12。证明对于任何整数m,任何正整数的平方不能为6m + 2或6m + 5的形式。
解决方案:
a = 6q + r, where 0 ≤ r < 6 (Taking b=6 in Euclid’s division lemma)
a2 = (6q + r)2 = 36q2 + r2 + 12qr
a2 = 6(6q2 + 2qr) + r2 0 ≤ r < 6
r = 0
a2 = 6 (6q2) = 6m, where, m = 6q2 is an integer.
r = 1
a2 = 6 (6q2 + 2q) + 1 = 6m + 1, where, m = (6q2 + 2q) is an integer.
r = 2,
a2 = 6(6q2 + 4q) + 4 = 6m + 4, where, m = (6q2 + 4q) is an integer.
r = 3,
a2 = 6(6q2 + 6q) + 9 = 6(6q2 + 6q) + 6 + 3
a2 = 6(6q2 + 6q + 1) + 3 = 6m + 3, where, m = (6q + 6q + 1) is integer.
r = 4,
a2 = 6(6q2 + 8q) + 16
= 6(6q2 + 8q) + 12 + 4
⇒ a2 = 6(6q2 + 8q + 2) + 4 = 6m + 4, where, m = (6q2 + 8q + 2) is integer.
r = 5,
a2 = 6 (6q2 + 10q) + 25 = 6(6q2 + 10q) + 24 + 1
a2 = 6(6q2 + 10q + 4) + 1 = 6m + 1, where, m = (6q2 + 10q + 4) is integer.
Therefore, the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.
问题13。证明正整数为6q + r,q为整数且r = 0、1、2、3、4、5的立方也为6m + r。
解决方案:
For 6q,
(6q)3 = 216 q3 = 6(36q)3 + 0
= 6m + 0, (where m is an integer = (36q)3)
For 6q+1,
(6q+1)3 = 216q3 + 108q2 + 18q + 1
= 6(36q3 + 18q2 + 3q) + 1
= 6m + 1, (where m is an integer = 36q3 + 18q2 + 3q)
For 6q+2,
(6q+2)3 = 216q3 + 216q2 + 72q + 8
= 6(36q3 + 36q2 + 12q + 1) +2
= 6m + 2, (where m is an integer = 36q3 + 36q2 + 12q + 1)
For 6q+3,
(6q+3)3 = 216q3 + 324q2 + 162q + 27
= 6(36q3 + 54q2 + 27q + 4) + 3
= 6m + 3, (where m is an integer = 36q3 + 54q2 + 27q + 4)
For 6q+4,
(6q+4)3 = 216q3 + 432q2 + 288q + 64
= 6(36q3 + 72q2 + 48q + 10) + 4
= 6m + 4, (where m is an integer = 36q3 + 72q2 + 48q + 10)
For 6q+5,
(6q+5)3 = 216q3 + 540q2 + 450q + 125
= 6(36q3 + 90q2 + 75q + 20) + 5
= 6m + 5, (where m is an integer = 36q3 + 90q2 + 75q + 20)
Therefore, the cube of a positive integer of the form 6q + r, q is an integer and r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r.
问题14.证明n,n + 4,n + 8,n + 12和n + 16中只有一个可以被5整除,其中n是任何正整数。
解决方案:
b=5
n = 5q+r
0 < r < 5
Therefore, n may be in the form of 5q, 5q+1, 5q+2, 5q+3, 5q+4
CASE 1:
When, n = 5q
n+4 = 5q+4
n+8 = 5q+8
n+12 = 5q+12
n+16 = 5q+16
n is only divisible by 5
CASE 2:
n = 5q+1
n+4 = 5q+5 = 5(q+1)
n+8 = 5q+9
n+12 = 5q+13
n+16 = 5q+17
n + 4 is only divisible by 5
CASE 3:
n = 5q+2
n+4 = 5q+6
n+8 = 5q+10 = 5(q+2)
n+12 = 5q+14
n+16 = 5q+18
n + 8 is only divisible by 5
CASE 4:
n = 5q+3
n+4 = 5q+7
n+8 = 5q+11
n+12 = 5q+15 = 5(q+3)
n+16 = 5q+19
n + 12 is only divisible by 5
CASE 5:
n = 5q+4
n+4 = 5q+8
n+8 = 5q+12
n+12 = 5q+16
n+16 = 5q+20 = 5(q+4)
Here, n + 16 is only divisible by 5
Therefore, one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5.
问题15:对于一个整数q,证明一个奇数整数的平方可以是6q +1或6q + 3的形式。
解决方案:
b=6
a = 6m + r
0 ≤ r < 6.
a = 6m, 6m + 1, 6m + 2 , 6m + 3, 6m + 4, 6m + 5
Thus, we are choosing for a = 6m + 1 or, 6m + 3 or 6m + 5 for it to be an odd integer.
For a = 6m + 1,
(6m + 1)2 = 36m2 + 12m + 1
= 6(6m2 + 2m) + 1
= 6q + 1, where q is some integer and q = 6m2 + 2m.
For a = 6m + 3
(6m + 3)2 = 36m2 + 36m + 9
= 6(6m2 + 6m + 1) + 3
= 6q + 3, where q is some integer and q = 6m2 + 6m + 1
For a = 6m + 5,
(6m + 5)2 = 36m2 + 60m + 25
= 6(6m2 + 10m + 4) + 1
= 6q + 1, where q is some integer and q = 6m2 + 10m + 4.
Therefore, the square of an odd integer is of the form 6q + 1 or 6q + 3, for some integer q.
问题16.正整数的形式为3q + 1,q是自然数。您是否可以将3m + 1、3m或3m + 2以外的任何形式的正方形写成某个整数m?证明你的答案。
解决方案:
No.
a = bq + r, 0 ≤ r < b
Here, a is any positive integer and b = 3,
⇒ a = 3q + r
So, a can be of the form 3q, 3q + 1 or 3q + 2.
Now, for a = 3q
(3q)2 = 3(3q2) = 3m [where m = 3q2]
a = 3q + 1
(3q + 1)2 = 9q2 + 6q + 1 = 3(3q2 + 2q) + 1 = 3m + 1 [where m = 3q2 + 2q]
a = 3q + 2
(3q + 2)2 = 9q2 + 12q + 4 = 9q2 + 12q + 3 + 1 = 3(3q2 + 4q + 1) + 1
= 3m + 1 [where m = 3q2 + 4q + 1]
Therefore, square of a positive integer of the form 3q + 1 is always of the form 3m + 1 or 3m for some integer m.
问题17.证明任何正整数的平方不能为3m + 2的形式,其中m是自然数。
解决方案:
a = bm + r
b = 3
a = 3m + r
r = 0, 1, 2.
r = 0, a = 3m.
r = 1, a = 3m + 1.
r = 2, a = 3m + 2.
When a = 3m
a2 = (3m)2 = 9m2
a2 = 3(3m2) = 3q, where q = 3m2
When a = 3m + 1
a2 = (3m + 1)2 = 9m2 + 6m + 1
a2 = 3(3m2 + 2m) + 1 = 3q + 1, where q = 3m2 + 2m
When a = 3m + 2
a2 = (3m + 2)2
a2 = 9m2 + 12m + 4
a2 = 3(3m2 + 4m + 1) + 1
a2 = 3q + 1 where q = 3m2 + 4m + 1
Therefore, square of any positive integer cannot be of the form 3q + 2, where q is a natural number.