Number Play uses algebra, visualisation and reasoning to explore parity, divisibility and number puzzles. Beginning with sums of consecutive numbers and the parities of expressions \(a \pm b \pm c \pm d\), it shows all such combinations share the same parity. Algebra explains when expressions are always even — for example \(4m + 2q = 2(2m+q)\) is always even. The chapter develops general divisibility rules and proves why they work: divisibility by \(2, 5, 10\) from the units digit; by \(9\) and \(3\) from the digit sum (since each place value is one more than a multiple of \(9\)); and by \(11\) from the alternating digit sum (since place values alternate one more and one less than a multiple of \(11\)). Key divisibility laws are established: if \(a\) divides \(M\) and \(N\), it divides \(M+N\) and \(M-N\); if \(a\) is divisible by \(k\) then so are all multiples of \(a\); and if a number is divisible by both \(k\) and \(m\), it is divisible by their LCM. Digital roots and cryptarithms — puzzles where letters stand for digits — round out the playful but rigorous tour.
Parity of expressionsMultiples and divisibility lawsDivisibility rules for 3, 9 and 11Digital rootsCryptarithms
Key terms
Parity
Whether a number is even or odd; sums and differences follow rules like even \(\pm\) even \(=\) even.
Divisibility rule
A shortcut to test if a number is divisible by another, such as the digit-sum test for \(9\).
Digit sum
The sum of all digits of a number; if it is divisible by \(9\) (or \(3\)), so is the number.
Alternating sum
The result of alternately adding and subtracting digits, used to test divisibility by \(11\).
Digital root
The single digit reached by repeatedly adding a number’s digits; it equals the remainder on dividing by \(9\).
Cryptarithm
A puzzle in which each letter represents a distinct digit and the first digit is never \(0\).
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Why is \(4m + 2q\) always even?
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