diff --git a/Mathematics/Prealgebra/Add_Subtract_Fractions.md b/Mathematics/Prealgebra/Add_Subtract_Fractions.md
index 8234a53..ce45739 100644
--- a/Mathematics/Prealgebra/Add_Subtract_Fractions.md
+++ b/Mathematics/Prealgebra/Add_Subtract_Fractions.md
@@ -36,6 +36,94 @@ $$
 * Add/subtract
 * Reduce 
 
-Demonstration: $\frac{4}{9} + \frac{1}{6}$ :
 
-The lowest common denominator is **the smallest number divisible by both of the denominators of the fractions without remainder**.
+
+### Lowest common denominator and lowest common multiple
+
+Given the symmetry between [factors and divisors](/Mathematics/Prealgebra/Factors_and_divisors.md) these properties are related. Note however that the LCM is more generic: it applies to any set of numbers not just fractions. Whereas the LCD is explicitly to do with fractions (hence 'denominator').
+
+
+* For two fractions $a, b$ (or a set), the LCD is the smallest number divisble by both the denominator of $a$ and the denominator of $b$ (or each member of the set).
+
+
+* For two fractions $a, b$ (or a set), the LCM is the smallest number that is a multiple of the denominator of $a$ and the denominator of $b$ (or each member of the set).
+
+In order to find the LCM of the set $\{12, 16\}$ we list the multiples of both:
+$$
+12, 24, 36, 48 \\
+16, 32, 48
+$$
+
+Until we identify the smallest number common to both lists. In this case it is 48. Thus the LCM of 12 and 16 is 48.
+
+The relationship between LCM and LCD is that _the least common denominator is the least common multiple of the fractions' denomintors_.  
+
+
+
+### Demonstration: addition
+We can now use this to calculate the addition of two fractions without common denominators: $\frac{4}{9} + \frac{1}{6}$.
+
+First identify the common multiples of 9 and 6:
+
+$$
+9, 18, ... \\
+6, 12, 18, ...
+$$
+
+The least common multiple is 18.
+We then think: what do we need to multiply each denominator by to get 18?
+
+In the case of the first fraction ($\frac{4}{9}$) it is 2:
+
+$$
+    \frac{4}{9 \cdot 2}  = \frac{4}{18}
+$$
+
+But what we do to the denominator, we must also do to the numerator, hence: 
+
+$$
+    \frac{4 \cdot 2}{9 \cdot 2}  = \frac{8}{18}
+$$
+
+We then do the same to the second fraction ($\frac{1}{6}$). We need to multiply its denominator by 3 to get 18 and we apply this also to the numerator.
+
+$$
+    \frac{1 \cdot 3}{6 \cdot 3}  = \frac{3}{18}
+$$
+
+We now have two fractions that share a common denominator so we can sum:
+
+$$
+    \frac{8}{18} + \frac{3}{18} = \frac{11}{18}
+$$
+
+### Demonstration: subtraction
+
+Calculate: 
+
+$$
+    \frac{3}{5} - \frac{2}{3}
+$$
+
+Once again we need to find the least common denominator for the two fractions. We start by listing the common multiples for the two denominators 5 and 3:
+
+$$
+5, 10, 15, ... \\
+3, 6, 9, 12, 15,...
+$$
+
+The lowest common multiple is 15. From the first fraction we get 15 by multiplying by 3. With the second fraction we get 15 by multiplying by 5. Thus:
+
+$$
+    \frac{3 \cdot 3}{5 \cdot 3}  = \frac{9}{15}
+$$
+
+$$
+    \frac{2 \cdot 5}{3 \cdot 5}  = \frac{10}{15}
+$$
+
+We can now carry out the subtraction:
+
+$$
+    \frac{9}{15}  -  \frac{10}{15} = -\frac{1}{15}
+$$
diff --git a/Mathematics/Prealgebra/Least_common_denominator.md b/Mathematics/Prealgebra/Least_common_denominator.md
deleted file mode 100644
index e69de29..0000000
diff --git a/pdf/Algorithmic_complexity.pdf b/pdf/Algorithmic_complexity.pdf
index 3cb41f0..4bbe665 100644
Binary files a/pdf/Algorithmic_complexity.pdf and b/pdf/Algorithmic_complexity.pdf differ
diff --git a/pdf/Arrays.pdf b/pdf/Arrays.pdf
index 55bc46f..f4c5364 100644
Binary files a/pdf/Arrays.pdf and b/pdf/Arrays.pdf differ
diff --git a/pdf/Assembly_programs.pdf b/pdf/Assembly_programs.pdf
index 43631a7..8048f49 100644
Binary files a/pdf/Assembly_programs.pdf and b/pdf/Assembly_programs.pdf differ
diff --git a/pdf/Atomic_and_molecular_sentences.pdf b/pdf/Atomic_and_molecular_sentences.pdf
index 2d98119..19a2bbe 100644
Binary files a/pdf/Atomic_and_molecular_sentences.pdf and b/pdf/Atomic_and_molecular_sentences.pdf differ
diff --git a/pdf/Axioms_of_set_theory.pdf b/pdf/Axioms_of_set_theory.pdf
index b6c1b7f..39bf772 100644
Binary files a/pdf/Axioms_of_set_theory.pdf and b/pdf/Axioms_of_set_theory.pdf differ
diff --git a/pdf/Basic_database_concepts.pdf b/pdf/Basic_database_concepts.pdf
index 258b957..87a8bb6 100644
Binary files a/pdf/Basic_database_concepts.pdf and b/pdf/Basic_database_concepts.pdf differ
diff --git a/pdf/Basic_properties_of_sets.pdf b/pdf/Basic_properties_of_sets.pdf
index 858f3bd..a7b6fbc 100644
Binary files a/pdf/Basic_properties_of_sets.pdf and b/pdf/Basic_properties_of_sets.pdf differ
diff --git a/pdf/Biconditional_Elimination.pdf b/pdf/Biconditional_Elimination.pdf
index 75522d3..16e57be 100644
Binary files a/pdf/Biconditional_Elimination.pdf and b/pdf/Biconditional_Elimination.pdf differ
diff --git a/pdf/Biconditional_Introduction.pdf b/pdf/Biconditional_Introduction.pdf
index eaf8bae..5f6345b 100644
Binary files a/pdf/Biconditional_Introduction.pdf and b/pdf/Biconditional_Introduction.pdf differ
diff --git a/pdf/Binary_arithmetic.pdf b/pdf/Binary_arithmetic.pdf
index f507bac..e36e48f 100644
Binary files a/pdf/Binary_arithmetic.pdf and b/pdf/Binary_arithmetic.pdf differ
diff --git a/pdf/Binary_arithmetic_with_circuits.pdf b/pdf/Binary_arithmetic_with_circuits.pdf
index 3ac5ca0..4e0c1d7 100644
Binary files a/pdf/Binary_arithmetic_with_circuits.pdf and b/pdf/Binary_arithmetic_with_circuits.pdf differ
diff --git a/pdf/Bits,_bytes,_nibbles.pdf b/pdf/Bits,_bytes,_nibbles.pdf
index ccb4b97..5aa1beb 100644
Binary files a/pdf/Bits,_bytes,_nibbles.pdf and b/pdf/Bits,_bytes,_nibbles.pdf differ
diff --git a/pdf/Compile_from_source.pdf b/pdf/Compile_from_source.pdf
index 50b17b3..8fabdd3 100644
Binary files a/pdf/Compile_from_source.pdf and b/pdf/Compile_from_source.pdf differ
diff --git a/pdf/Conditional_Elimination.pdf b/pdf/Conditional_Elimination.pdf
index ddec612..da107c3 100644
Binary files a/pdf/Conditional_Elimination.pdf and b/pdf/Conditional_Elimination.pdf differ
diff --git a/pdf/Conditional_Introduction.pdf b/pdf/Conditional_Introduction.pdf
index 85a1797..0e0c2ee 100644
Binary files a/pdf/Conditional_Introduction.pdf and b/pdf/Conditional_Introduction.pdf differ
diff --git a/pdf/Conjunction_Elimination.pdf b/pdf/Conjunction_Elimination.pdf
index 6478e4c..004fb4d 100644
Binary files a/pdf/Conjunction_Elimination.pdf and b/pdf/Conjunction_Elimination.pdf differ
diff --git a/pdf/Conjunction_Introduction.pdf b/pdf/Conjunction_Introduction.pdf
index d9ef8c2..a99ce75 100644
Binary files a/pdf/Conjunction_Introduction.pdf and b/pdf/Conjunction_Introduction.pdf differ
diff --git a/pdf/Defining_a_computer.pdf b/pdf/Defining_a_computer.pdf
index f02299f..9ce2122 100644
Binary files a/pdf/Defining_a_computer.pdf and b/pdf/Defining_a_computer.pdf differ
diff --git a/pdf/Design_features_of_language.pdf b/pdf/Design_features_of_language.pdf
index 0fdaea3..af4884e 100644
Binary files a/pdf/Design_features_of_language.pdf and b/pdf/Design_features_of_language.pdf differ
diff --git a/pdf/Disjunction_Elimination.pdf b/pdf/Disjunction_Elimination.pdf
index 52a34f6..827cf95 100644
Binary files a/pdf/Disjunction_Elimination.pdf and b/pdf/Disjunction_Elimination.pdf differ
diff --git a/pdf/Disjunction_Introduction.pdf b/pdf/Disjunction_Introduction.pdf
index e1a50d2..26d10c7 100644
Binary files a/pdf/Disjunction_Introduction.pdf and b/pdf/Disjunction_Introduction.pdf differ
diff --git a/pdf/HTTP_request_types.pdf b/pdf/HTTP_request_types.pdf
index 62f81e8..49c680c 100644
Binary files a/pdf/HTTP_request_types.pdf and b/pdf/HTTP_request_types.pdf differ
diff --git a/pdf/Indeterminacy.pdf b/pdf/Indeterminacy.pdf
index 6082265..ea127ab 100644
Binary files a/pdf/Indeterminacy.pdf and b/pdf/Indeterminacy.pdf differ
diff --git a/pdf/Law_of_the_Excluded_Middle.pdf b/pdf/Law_of_the_Excluded_Middle.pdf
index 6075aba..9cdc283 100644
Binary files a/pdf/Law_of_the_Excluded_Middle.pdf and b/pdf/Law_of_the_Excluded_Middle.pdf differ
diff --git a/pdf/Logic_circuits.pdf b/pdf/Logic_circuits.pdf
index 316c97d..e9a19e6 100644
Binary files a/pdf/Logic_circuits.pdf and b/pdf/Logic_circuits.pdf differ
diff --git a/pdf/Logic_gates.pdf b/pdf/Logic_gates.pdf
index fc7c736..2b98a16 100644
Binary files a/pdf/Logic_gates.pdf and b/pdf/Logic_gates.pdf differ
diff --git a/pdf/Logical_possibility_and_necessity.pdf b/pdf/Logical_possibility_and_necessity.pdf
index 20b6abe..dbed976 100644
Binary files a/pdf/Logical_possibility_and_necessity.pdf and b/pdf/Logical_possibility_and_necessity.pdf differ
diff --git a/pdf/Lovelace_quote.pdf b/pdf/Lovelace_quote.pdf
index d2647dd..d81361c 100644
Binary files a/pdf/Lovelace_quote.pdf and b/pdf/Lovelace_quote.pdf differ
diff --git a/pdf/Morphology.pdf b/pdf/Morphology.pdf
index a1b1759..442a666 100644
Binary files a/pdf/Morphology.pdf and b/pdf/Morphology.pdf differ
diff --git a/pdf/Negation_Elimination.pdf b/pdf/Negation_Elimination.pdf
index 6731a89..45f6fa4 100644
Binary files a/pdf/Negation_Elimination.pdf and b/pdf/Negation_Elimination.pdf differ
diff --git a/pdf/Negation_Introduction.pdf b/pdf/Negation_Introduction.pdf
index ffaece5..ff3b905 100644
Binary files a/pdf/Negation_Introduction.pdf and b/pdf/Negation_Introduction.pdf differ
diff --git a/pdf/Object_language_and_meta-language.pdf b/pdf/Object_language_and_meta-language.pdf
index f4d0c3a..7e96509 100644
Binary files a/pdf/Object_language_and_meta-language.pdf and b/pdf/Object_language_and_meta-language.pdf differ
diff --git a/pdf/Package_management_in_Arch.pdf b/pdf/Package_management_in_Arch.pdf
index cb6c27a..56b0cbb 100644
Binary files a/pdf/Package_management_in_Arch.pdf and b/pdf/Package_management_in_Arch.pdf differ
diff --git a/pdf/Primary_key.pdf b/pdf/Primary_key.pdf
index f88ba84..b8d561f 100644
Binary files a/pdf/Primary_key.pdf and b/pdf/Primary_key.pdf differ
diff --git a/pdf/Queue.pdf b/pdf/Queue.pdf
index ae979cc..94608b6 100644
Binary files a/pdf/Queue.pdf and b/pdf/Queue.pdf differ
diff --git a/pdf/RESTful_APIs.pdf b/pdf/RESTful_APIs.pdf
index bbf94cf..5cedef8 100644
Binary files a/pdf/RESTful_APIs.pdf and b/pdf/RESTful_APIs.pdf differ
diff --git a/pdf/Recursion.pdf b/pdf/Recursion.pdf
index 2dc0cba..e84d0b7 100644
Binary files a/pdf/Recursion.pdf and b/pdf/Recursion.pdf differ
diff --git a/pdf/Reiteration.pdf b/pdf/Reiteration.pdf
index 000bf73..e13810c 100644
Binary files a/pdf/Reiteration.pdf and b/pdf/Reiteration.pdf differ
diff --git a/pdf/SQL_syntax.pdf b/pdf/SQL_syntax.pdf
index ce550da..3ac26b9 100644
Binary files a/pdf/SQL_syntax.pdf and b/pdf/SQL_syntax.pdf differ
diff --git a/pdf/Semantic_versioning.pdf b/pdf/Semantic_versioning.pdf
index ce6a239..bc15df6 100644
Binary files a/pdf/Semantic_versioning.pdf and b/pdf/Semantic_versioning.pdf differ
diff --git a/pdf/Soundness.pdf b/pdf/Soundness.pdf
index 72f7052..d2e6e83 100644
Binary files a/pdf/Soundness.pdf and b/pdf/Soundness.pdf differ
diff --git a/pdf/Stacks.pdf b/pdf/Stacks.pdf
index 939f032..4f9e172 100644
Binary files a/pdf/Stacks.pdf and b/pdf/Stacks.pdf differ
diff --git a/pdf/Syllogism.pdf b/pdf/Syllogism.pdf
index 0c63c18..e2fd8f8 100644
Binary files a/pdf/Syllogism.pdf and b/pdf/Syllogism.pdf differ
diff --git a/pdf/The_Pragmatic_Programmer_1999.pdf b/pdf/The_Pragmatic_Programmer_1999.pdf
index 0e10020..26a9c5e 100644
Binary files a/pdf/The_Pragmatic_Programmer_1999.pdf and b/pdf/The_Pragmatic_Programmer_1999.pdf differ
diff --git a/pdf/The_binary_number_system.pdf b/pdf/The_binary_number_system.pdf
index 4b9130b..8ebcfec 100644
Binary files a/pdf/The_binary_number_system.pdf and b/pdf/The_binary_number_system.pdf differ
diff --git a/pdf/Theorems_and_empty_sets.pdf b/pdf/Theorems_and_empty_sets.pdf
index 2e3a390..2ca1aeb 100644
Binary files a/pdf/Theorems_and_empty_sets.pdf and b/pdf/Theorems_and_empty_sets.pdf differ
diff --git a/pdf/Turing_machines.pdf b/pdf/Turing_machines.pdf
index 4108017..7a983f2 100644
Binary files a/pdf/Turing_machines.pdf and b/pdf/Turing_machines.pdf differ
diff --git a/pdf/User_management.pdf b/pdf/User_management.pdf
index b65b0dd..08939db 100644
Binary files a/pdf/User_management.pdf and b/pdf/User_management.pdf differ
diff --git a/pdf/Von_Neumann_architecture.pdf b/pdf/Von_Neumann_architecture.pdf
index 55d8128..4171548 100644
Binary files a/pdf/Von_Neumann_architecture.pdf and b/pdf/Von_Neumann_architecture.pdf differ
diff --git a/pdf/Why_computers_use_binary.pdf b/pdf/Why_computers_use_binary.pdf
index d5d2c09..30435ee 100644
Binary files a/pdf/Why_computers_use_binary.pdf and b/pdf/Why_computers_use_binary.pdf differ