Let "MTar" mean a Modulus-Target pair.  So I can describe the congruence

	x =42= 34

by the MTar (42, 34).  


================================================================
  FIRST EXAMPLE:	(fuse-many-MTars '(42 34) '(27 12))
Our goal is to solve this system of congruences:

C1:  x =42= 34.     C2:  x =27= 12.     

/--------------------------------------------------------\
With A:=42 and B:=27, is there an x with
  x =A=  34 and  |  Alas, NO.  This, since
  x =B=  12 ?    |  Gcd(A,B)=3 does not divide 34-12 .
\________________________________________________________/




================================================================
  SECOND EXAMPLE:	(fuse-many-MTars '(12 7) '(21 1) '(35 22))
Our goal is to solve this system of congruences:

C1:  x =12=  7.     C2:  x =21=  1.     C3:  x =35= 22.

/--------------------------------------------------------\
With A:=12 and B:=21, is there an x with
  x =A=   7 and  |  YES, since  D := Gcd(A,B)=3 divides 7-1 =: F.
  x =B=   1 ?    |  Their ratio is  R := F/D = 2.

LBolt gives 2A + -1B = D.  So  4A - 2B = R*D  =  7 - 1.
  Thus  x + 4A  =  7.  So  x := -41  solves the two given congruences.
  Reducing this modulo  L := Lcm(A,B)=84 gives x := [-41 mod L] = 43.

Upshot: We can fuse the two  (Modulus Target) pairs into a single (M T):
  (12 7) fuse (21 1)   =   (L x)   =note=   (84 43) .
\________________________________________________________/

/--------------------------------------------------------\
With A:=84 and B:=35, is there an x with
  x =A=  43 and  |  YES, since  D := Gcd(A,B)=7 divides 43-22 =: F.
  x =B=  22 ?    |  Their ratio is  R := F/D = 3.

LBolt gives -2A + 5B = D.  So  -6A - -15B = R*D  =  43 - 22.
  Thus  x + -6A  =  43.  So  x := 547  solves the two given congruences.
  Reducing this modulo  L := Lcm(A,B)=420 gives x := [547 mod L] = 127.

Upshot: We can fuse the two  (Modulus Target) pairs into a single (M T):
  (84 43) fuse (35 22)   =   (L x)   =note=   (420 127) .
\________________________________________________________/



================================================================
  THIRD EXAMPLE: Here, the first fusion succeeds but the second fusion fails.

			(fuse-many-MTars '(18 1) '(20 7) '(21 2))
Our goal is to solve this system of congruences:

C1:  x =18=  1.     C2:  x =20=  7.     C3:  x =21=  2.

/--------------------------------------------------------\
With A:=18 and B:=20, is there an x with
  x =A=   1 and  |  YES, since  D := Gcd(A,B)=2 divides 1-7 =: F.
  x =B=   7 ?    |  Their ratio is  R := F/D = -3.

LBolt gives -1A + 1B = D.  So  3A - 3B = R*D  =  1 - 7.
  Thus  x + 3A  =  1.  So  x := -53  solves the two given congruences.
  Reducing this modulo  L := Lcm(A,B)=180 gives x := [-53 mod L] = 127.

Upshot: We can fuse the two  (Modulus Target) pairs into a single (M T):
  (18 1) fuse (20 7)   =   (L x)   =note=   (180 127) .
\________________________________________________________/

/--------------------------------------------------------\
With A:=180 and B:=21, is there an x with
  x =A= 127 and  |  Alas, NO.  This, since
  x =B=   2 ?    |  Gcd(A,B)=3 does not divide 127-2 .
\________________________________________________________/

================================================================
  FOURTH EXAMPLE: Here, the target value changes after each fusion.

		(fuse-many-MTars '(22 5) '(14 11) '(21 18) '(15 3))
Our goal is to solve this system of congruences:

C1:  x =22=  5.    C2:  x =14= 11.    C3:  x =21= 18.   C4:  x =15=  3.

/--------------------------------------------------------\
With A:=22 and B:=14, is there an x with
  x =A=   5 and  |  YES, since  D := Gcd(A,B)=2 divides 5-11 =: F.
  x =B=  11 ?    |  Their ratio is  R := F/D = -3.

LBolt gives 2A + -3B = D.  So  -6A - -9B = R*D  =  5 - 11.
  Thus  x + -6A  =  5.  So  x := 137  solves the two given congruences.
  Reducing this modulo  L := Lcm(A,B)=154 gives x := [137 mod L] = 137.

Upshot: We can fuse the two  (Modulus Target) pairs into a single (M T):
  (22 5) fuse (14 11)   =   (L x)   =note=   (154 137) .
\________________________________________________________/

/--------------------------------------------------------\
With A:=154 and B:=21, is there an x with
  x =A= 137 and  |  YES, since  D := Gcd(A,B)=7 divides 137-18 =: F.
  x =B=  18 ?    |  Their ratio is  R := F/D = 17.

LBolt gives 1A + -7B = D.  So  17A - 119B = R*D  =  137 - 18.
  Thus  x + 17A  =  137.  So  x := -2481  solves the two given congruences.
  Reducing this modulo  L := Lcm(A,B)=462 gives x := [-2481 mod L] = 291.

Upshot: We can fuse the two  (Modulus Target) pairs into a single (M T):
  (154 137) fuse (21 18)   =   (L x)   =note=   (462 291) .
\________________________________________________________/

/--------------------------------------------------------\
With A:=462 and B:=15, is there an x with
  x =A= 291 and  |  YES, since  D := Gcd(A,B)=3 divides 291-3 =: F.
  x =B=   3 ?    |  Their ratio is  R := F/D = 96.

LBolt gives -1A + 31B = D.  So  -96A - -2976B = R*D  =  291 - 3.
  Thus  x + -96A  =  291.  So  x := 44643  solves the two given congruences.
  Reducing this modulo  L := Lcm(A,B)=2310 gives x := [44643 mod L] = 753.

Upshot: We can fuse the two  (Modulus Target) pairs into a single (M T):
  (462 291) fuse (15 3)   =   (L x)   =note=   (2310 753) .
\________________________________________________________/


;;;; End of "fuse-example.txt" ;;;;