List of questions

1. What is the smallest positive integer > 1 that has a

remainder of 1, when divided by each of 3,5 and 7?

2. How can I find a set of four integers such that the sum

of any two of them is a square.

Question # 1

What is the smallest positive integer > 1 that has a

remainder of 1, when divided by each of 3,5 and 7?

Let x be the name of the integer we are to find.

If x has a remainder of 1 when divided by 3,

then (x-1) has a remainder of 0 when divided by 3.

That is, (x-1) is a multiple of 3.

If x has a remainder of 1 when divided by 5,

then (x-1) is a multiple of 5.

If x has a remainder of 1 when divided by 7,

then (x-1) is a multiple of 7.

The smallest positive integer that is, at the same time,

a multiple of 3, a multiple of 5, and a multiple of 7,

is 3 * 5 * 7 = 105.

(x-1) = 105

x = 106.

106 is the smallest positive integer > 1 such that

the remainder is 1 when divided by any of 3,5, or 7.

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Question # 2

How can I find a set of four integers such that the sum of

any two of them is a square.

Choose 6 integers, t1,t2,t3,s1,s2,s3.

Define

a1 = real part of

((t1+s1+1)+i(t1-s1))((t2+s2+1)+i(t2-s2))((t3+s3+1)+i(t3-s3))

a2 = imaginary part of

((t1+s1+1)+i(t1-s1))((t2+s2+1)+i(t2-s2))((t3+s3+1)+i(t3-s3))

a3 = real part of

((t1+s1+1)+i(t1-s1))((t2+s2+1)+i(t2-s2))((t3+s3+1)-i(t3-s3))

a4 = imaginary part of

((t1+s1+1)+i(t1-s1))((t2+s2+1)+i(t2-s2))((t3+s3+1)-i(t3-s3))

a5 = real part of

((t1+s1+1)+i(t1-s1))((t2+s2+1)-i(t2-s2))((t3+s3+1)+i(t3-s3))

a6 = imaginary part of

((t1+s1+1)+i(t1-s1))((t2+s2+1)-i(t2-s2))((t3+s3+1)+i(t3-s3))

Define either

p1 = (a1**2 + a3**2 - a6**2)/2

p2 = (a1**2 + a6**2 - a3**2)/2

p3 = (a3**2 + a6**2 - a1**2)/2

p4 = (a5**2 + a4**2 - a1**2)/2

or

p1 = (a1**2 + a3**2 - a5**2)/2

p2 = (a1**2 + a5**2 - a3**2)/2

p3 = (a3**2 + a5**2 - a1**2)/2

p4 = (a6**2 + a4**2 - a1**2)/2

according to which will make p1,p2,p3,p4 to be integers.

p1 + p2 = a1**2

p1 + p3 = a3**2

p1 + p4 = a5**2

p2 + p3 = a6**2

p2 + p4 = a4**2

p3 + p4 = a2**2

or

p1 + p2 = a1**2

p1 + p3 = a3**2

p1 + p4 = a6**2

p2 + p3 = a5**2

p2 + p4 = a4**2

p3 + p4 = a2**2

Numerical Example

Choose t1 = 2, t2 = 3, t3 = 5, s1=1, s2=1,s3=1

(4+i)(5+2i)(7+4i)

= ((4+i)(5+2i))(7+4i)

= (18+13i)(7+4i)

= (74+163i)

a1 = 74

a2 = 163

(4+i)(5+2i)(7-4i)

= ((4+i)(5+2i))(7-4i)

= (18+13i)(7-4i)

= (178+19i)

a3 = 178

a4 = 19

(4+i)(5-2i)(7+4i)

= ((4+i)(5-2i))(7+4i)

= (22-3i)(7+4i)

= (166+67i)

a5 = 166

a6 = 67

a1**2 = 74**2 = 5476

a2**2 = 163**2 = 26569

a3**2 = 178**2 = 31684

a4**2 = 19**2 = 361

a5**2 = 166**2 = 27556

a6**2 = 67**2 = 4489

p1 = (a1**2 + a3**2 - a5**2)/2

p1 = (5476 + 31684 - 27556)/2

p1 = 4802

p2 = (a1**2 + a5**2 - a3**2)/2

p2 = (5476 + 27556 - 31684)/2

p2 = 674

p3 = (a3**2 + a5**2 - a1**2)/2

p3 = (31684 + 27556 - 5476)/2

p3 = 26882

p4 = (a6**2 + a4**2 - a1**2)/2

p4 = (4489 + 361 - 5476 )/2

p4 = -313

p1 + p2 = 5476 = 74**2

p1 + p3 = 31684 = 178**2

p1 + p4 = 4489 = 67**2

p2 + p3 = 27556 = 166**2

p2 + p4 = 361 = 19**2

p3 + p4 = 26569 = 163**2

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