Number properties and remainders

01 / UNDERSTAND & APPLY

Domains, signs and magnitude

In this lesson, natural numbers mean 1,2,3,… and whole numbers include 0. Some books use a different convention for natural numbers, so state it when zero matters. Integers include negative whole numbers. Rational numbers can be written as p/q with integers p,q and q nonzero; irrational real numbers cannot. Together rational and irrational numbers form the real numbers.

A finite decimal is rational because it can be written over a power of 10; an eventually repeating decimal is also rational. Not every infinite decimal is irrational. A nonterminating, nonrepeating decimal expansion characterizes an irrational real number in base 10. Examples include √2, whereas √9=3 is rational.

Order and magnitude are different. On the number line,−7 is less than−2, but its absolute value 7 exceeds 2. Absolute value is distance from zero and is never negative. A large negative number in magnitude is not a larger signed value.

02 / UNDERSTAND & APPLY

Place value and scale

In base 10, each position represents a power of 10. Thus 345=3×100+4×10+5. Multiplying by 10 shifts place value; it does not merely append a zero to every kind of written number. For decimals,3.45×10=34.5.

Scientific notation writes a positive value as a×10^k with 1≤a<10. Multiplication and division combine coefficients and add or subtract exponents. The scale check often reveals a misplaced thousand or million before exact arithmetic is finished.

“Order of magnitude” is sometimes used informally for scale and sometimes for the nearest power of 10. State which convention you mean. Here an estimate such as “a few thousand” is a scale check, not a claim that every source uses one rounding convention for the phrase.

03 / UNDERSTAND & APPLY

Why the familiar tests work

An integer n is divisible by a nonzero integer d when n=dq for an integer q. Zero is divisible by every nonzero integer because 0=d×0. Division by zero is undefined; do not treat it as a special case of a divisibility test.

For divisibility by 2,5 or 10, only the last digit matters because every higher place is a multiple of 10, and 10 is divisible by 2 and 5. Even last digits give divisibility by 2;0 or 5 by 5;0 by 10. For 4, the last two digits suffice because 100 is divisible by 4. For 8, the last three suffice because 1000 is divisible by 8.

For 3 and 9,10 leaves remainder 1, so every power of 10 does too. A number has the same remainder as its digit sum modulo 3 or 9. For 11,10 leaves remainder−1, so alternating place values give an alternating digit sum; divisibility holds when that sum is a multiple of 11, including 0. For example,121 gives 1−2+1=0. These tests establish divisibility of integers, not arbitrary decimals without conversion.

04 / UNDERSTAND & APPLY

Prime factors and divisor choices

A prime integer exceeds 1 and has exactly two positive divisors:1 and itself. One is not prime, and 0 is not prime. Two is the only even prime, because every even integer greater than 2 has 2 as a proper divisor. Composite positive integers greater than 1 have additional positive divisors.

Every integer greater than 1 has a unique prime factorization apart from factor order. A positive divisor chooses an exponent from 0 up to the available exponent of each prime. If n=p^a q^b for distinct primes, its divisor count is(a+1)(b+1). The multiplication counts independent exponent choices; it does not add the primes themselves.

05 / UNDERSTAND & APPLY

Common factors, common multiples and wording

The greatest common divisor takes the minimum exponent of each shared prime factor; the least common multiple takes the maximum exponent needed to include every factor of both numbers. For two positive integers a,b, gcd(a,b)×lcm(a,b)=ab because each prime’s minimum and maximum exponents sum to its two original exponents. Do not apply that two-number identity unchanged to three numbers.

Interpret the object being maximized. The greatest number of identical groups using every red and blue bead must divide both colour totals; the GCD gives the number of groups. In a different problem asking the largest equal-sized single-colour packets, the same arithmetic can refer to packet size. Name the unit of the GCD result.

An LCM finds the first positive alignment when cycles share the same starting event. If cycles have different starting offsets, solve the remainder conditions instead of simply taking the LCM from zero.

06 / UNDERSTAND & APPLY

Use the allowed remainder range

For integer n and positive divisor d, Euclidean division writes n=dq+r with integer q and 0≤r<d. This convention also applies when n is negative. For−17 divided by 5, choosing q=−4 gives r=3; a negative leftover−2 is not an allowed Euclidean remainder.

If a and b have remainders r and s modulo d, their sum and product have the same remainders as r+s and rs after reducing again. Terms containing a factor d contribute no remainder. This lets you work with small numbers rather than huge products.

For example, numbers leaving remainders 4 and 5 modulo 7 have sum remainder 2 because 4+5=9≡2, and product remainder 6 because 20≡6. Division needs more care: one cannot generally divide remainders by an arbitrary number without checking invertibility, so this lesson does not use that shortcut.

07 / UNDERSTAND & APPLY

Parity and power cycles

Even integers have form 2 k; odd integers have form 2 k+1. Adding two odds gives an even number, multiplying two odds gives an odd, and adding an even to an odd stays odd. These restrictions can eliminate impossible answers without determining the exact integer.

Last digits are remainders modulo 10. Powers often cycle because each step multiplies the current remainder by the same base. List a complete repeating cycle and map the exponent carefully. If the exponent remainder is 0, choose the final position of the cycle, not a nonexistent zeroth entry.

08 / UNDERSTAND & APPLY

Check bounds and the requested domain

A numerical answer should satisfy the original domain and all constraints. Consecutive squares can locate a square root without decimal approximation:8²=64<80<81=9², so√80 lies strictly between 8 and 9. The smallest integer greater than it is 9.

For divisibility in an interval, find the relevant common multiple and inspect only its multiples within the endpoints. Inclusive and exclusive endpoints can change the answer. Also distinguish a greatest integer below a value from a least integer above it, especially when the value itself is an integer.

Finish by checking signs, units and whether zero is permitted. A technically correct factor calculation can answer the wrong question if it labels a group count as a group size or ignores an excluded boundary.