Chapter Summary and Practice

A number can be described in several useful ways: as a sum of neighbours, a collection of equal groups, a string of digits, or a product of factors. The best description depends on the question. If you want parity, you may only need odd and even. If you want divisibility by nine, a digit sum may be enough. If letters hide digits, place value and carries can reveal what a full calculation would conceal.

Use this page to rebuild the connections across the chapter. Read a rule, explain why it works in your own words, and try the accompanying example before revealing its solution. The aim is to choose a method for a reason, rather than to guess which shortcut to use.

Summary

Consecutive integers increase by one: n, n + 1, n + 2, … . Consecutive even integers increase by two. Algebra records the spacing without choosing a starting value. Four consecutive integers have sum 4n + 6; three consecutive even integers have sum 2n + (2n + 2) + (2n + 4) = 6(n + 1).

When writing a number as a consecutive sum, state the rules. With at least two positive terms, 15 has several representations, including 7 + 8 and 4 + 5 + 6, whereas 8 has none. Allowing negative integers changes the answer: the integers from −7 through 8 sum to 8 because opposite pairs cancel. Zero can be written −1 + 0 + 1.

Even ± even and odd ± odd are even; mixed-parity sums and differences are odd. A product is even if at least one factor is even, and a product of only odd factors is odd. A positive integer power has the parity of its base. To show an expression is always even, expose a factor of 2, as in 4m + 2n = 2(2m + n).

Replacing +b by −b changes a total by −2b, so it preserves parity. Every choice of signs for a fixed list of integers has the same parity. For four consecutive integers this parity is even; the general results are 4n + 6, 2n, 2n + 2, −4, 2n + 4, −2, 0 and −2n − 6. The constant results −4, −2 and 0 occur whatever the starting integer.

Even integers are of two remainder types on division by 4: 4p and 4p + 2. Two type-0 numbers give remainder 0; two type-2 numbers give 2 + 2 = 4, also remainder 0. Mixing the types leaves remainder 2. Thus the sum is divisible by 4 exactly when the two even numbers have the same remainder type.

Always statements require an argument for every allowed input. Sometimes statements need both a successful example and a counterexample. Never statements need a contradiction or another reason excluding all cases. For example, an odd-plus-even sum can never be a multiple of 6 because the sum is odd but every multiple of 6 is even.

Combine remainders, then regroup if needed. For divisor 7, leftovers 5 and 3 add to 8, which becomes one full group plus remainder 1. Their difference is 2. If a signed difference is negative, add the divisor until it lies in the permitted range: −3 corresponds to remainder 4 for division by 7.

Equivalent descriptions may use different parameter values: 5k + 3 for k ≥ 0 and 5k − 2 for k ≥ 1 describe the same nonnegative sequence 3, 8, 13, … . To combine shared remainder conditions, subtract the remainder first. Remainder 2 for both 3 and 4 gives N − 2 divisible by 12, hence N = 12k + 2. One less than multiples of 3, 4 and 5 gives N = 60k − 1; the smallest positive answer is 59.

Place value is the foundation of digit tests. The string dcba denotes 1000d + 100c + 10b + a. For 3 and 9, each decimal place value is one more than a multiple of the divisor, so the digit sum preserves the remainder. For 11, place values alternate between one more and one less, producing the alternating-sign rule.

The last-two-digit test works because 100 is divisible by 4; the last-three-digit test works because 1000 is divisible by 8. Combining tests works when their LCM is the target divisor. Tests for 4 and 6 guarantee only 12, so they cannot establish divisibility by 24.

For 11, the signed result must be reduced before it is called a remainder. In 328105, the units-positive sum is −3, meaning three short of a multiple of 11; the standard remainder is −3 + 11 = 8. Zero, 11, −11 and 22 all indicate divisibility.

For positive integers, roots 1 through 8 match the remainder on division by 9; root 9 means remainder 0. For division by 3, roots 1, 4, 7 give remainder 1; roots 2, 5, 8 give remainder 2; roots 3, 6, 9 give remainder 0. A digital root does not determine parity: 18 and 27 both have root 9.

Digital roots of consecutive positive integers cycle through 1 to 9. Multiples of 3 cycle through 3, 6, 9; multiples of 4 through 4, 8, 3, 7, 2, 6, 1, 5, 9; multiples of 6 through 6, 3, 9. Adding 11 advances the remainder by 2 because 11 = 9 + 2. A positive expression 9a + 36b + 13 has root 4 because it equals 9(a + 4b + 1) + 4.

Translate letters by place value, inspect the strongest column, keep carries, and bound the number before trying digits. In A1 + 1B = B0, the units force B = 9 and carry 1; the tens then force A = 7. In PQ × 8 = RS, the two-digit product restricts PQ to 10, 11 or 12, and only 12 × 8 = 96 satisfies all letter rules.

Remember that some puzzles have several answers. L2N × 2 = 2NP works as 124 × 2 = 248 and 125 × 2 = 250. A successful assignment proves existence; checking the remaining possible digits proves completeness. If a restriction becomes impossible, a puzzle can also have no solution.

Multiples can also be compared as sets. Every multiple of 32 is a multiple of 8, and every multiple of 8 is a multiple of 4. The reverse statements fail: 12 is a multiple of 4 but not 8, and 24 is a multiple of 8 but not 32. So the set of multiples of 32 sits inside the set of multiples of 8, which sits inside the set of multiples of 4.

It’s Puzzle Time!

Navakankari

Navakankari is a two-player strategy game, also known by names including Sālu Mane Āṭa, Chār-Pār and Navkakri. Its board has three nested squares joined at the side midpoints. You try to form a connected horizontal or vertical line of three of your own pawns, then remove an opponent’s pawn. You can also win by leaving the opponent unable to move.

Draw the board below, or use the diagram as a guide. Give each player nine distinguishable pawns, such as two colours of counters. Pawns sit only on the marked points, with at most one pawn per point. The crossing at the centre is not a playing point: the middle of the board is open.

Gameplay

Placement: take turns placing one pawn on any empty marked point until both players have placed their nine pawns. Watch for pairs of your own pawns with an empty third point, and for an opponent’s pair that you may need to block.

Movement: after all pawns have been placed, take turns moving one of your pawns along a drawn line to an adjacent empty point. Adjacent means the next marked point along that line, without jumping over another point or pawn. Diagonal movement is not allowed.

Making a line: when you complete a connected horizontal or vertical line of three of your pawns, remove one opponent pawn that is not already part of one of their completed lines. A line must follow the board connections; three points that merely look aligned across the open centre do not make one line.

Winning: you win if your opponent has fewer than three pawns remaining or cannot make a legal move. The rules here use adjacent moves and do not add a flying move. Before playing, agree what to do if every opponent pawn is protected in a line, or if a position repeats indefinitely; those cases are not specified in these basic rules.