Number System: AFCAT Numerical Ability

Formulas (3)

• Rational or irrational, and terminating or repeating · Terminating test
  $$\dfrac pq \text{ in lowest terms terminates} \;\text{means}\; q = 2^a \times 5^b$$
• Primes, composites, even and odd numbers · Prime test
  $$n > 1 \text{ is prime when no prime } p \le \sqrt n \text{ divides it}$$
• Perfect squares and perfect cubes · Bracketing test
  $$n^3 < N < (n+1)^3 \;\Rightarrow\; N \text{ is not a perfect cube}$$

Reference tables (1)

Watch out for (9)

• Zero is whole, not natural
  → Natural, whole, integer, rational, irrational and real numbers
• The chain runs one way
  → Natural, whole, integer, rational, irrational and real numbers
• Simplify before you test
  → Rational or irrational, and terminating or repeating
• 22/7 is not pi
  → Rational or irrational, and terminating or repeating
• A long decimal can still repeat
  → Rational or irrational, and terminating or repeating
• 1 is not prime
  → Primes, composites, even and odd numbers
• Odd is not prime
  → Primes, composites, even and odd numbers
• A cube is not automatically a square
  → Perfect squares and perfect cubes
• Count the zeros
  → Perfect squares and perfect cubes

Formulas (5)

• Rules that add the digits: 3 and 9 · Digit-sum rule
  $$N \text{ divisible by } 9 \;\text{means}\; \text{its digit sum is divisible by } 9$$
• Rules for 7 and 11 · Rule for 11
  $$(d_0 + d_2 + d_4 + \cdots) - (d_1 + d_3 + \cdots) = 0, \pm 11, \pm 22, \ldots$$
• Divisors made of two rules: 6, 12, 15, 18, 36 · Combining two rules
  $$a \mid N,\ b \mid N,\ a \text{ and } b \text{ share no factor} \;\Rightarrow\; ab \mid N$$
• A missing digit, or the least number to add or take away · Least number to add or subtract
  $$N = dq + r:\quad \text{add } d - r,\ \text{or subtract } r$$
• Counting and adding the multiples in a range · Multiples of d from d·m to d·n
  $$\text{count} = n - m + 1,\qquad \text{sum} = \dfrac{\text{count} \times (dm + dn)}{2}$$

Reference tables (1)

Watch out for (8)

• Two digits are not enough for 8
  → Rules that read the last digits: 2, 4, 5, 8, 10 and 25
• Divisible by 3 is not divisible by 9
  → Rules that add the digits: 3 and 9
• 11 passes as well as 0
  → Rules for 7 and 11
• Subtract twice the last digit
  → Rules for 7 and 11
• The pieces must share nothing
  → Divisors made of two rules: 6, 12, 15, 18, 36
• Add the divisor minus the remainder
  → A missing digit, or the least number to add or take away
• Count both ends
  → Counting and adding the multiples in a range
• Both 4 and 6 means 12
  → Counting and adding the multiples in a range

Formulas (4)

• Writing a prime factorisation · Prime factorisation
  $$N = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k} \quad (p_i \text{ prime})$$
• Reading a factorisation: distinct primes, repeats and the number itself · For N = pᵃ × qᵇ × rᶜ
  $$\text{distinct primes} = 3,\qquad \text{primes with repeats} = a + b + c$$
• How many divisors a number has · Number of divisors
  $$N = p^a q^b r^c \;\Rightarrow\; d(N) = (a + 1)(b + 1)(c + 1)$$
• Odd divisors, even divisors and the sum of the divisors · Odd and even divisors of 2ᵃ × qᵇ × rᶜ
  $$\text{odd} = (b + 1)(c + 1),\qquad \text{even} = a(b + 1)(c + 1)$$

Watch out for (7)

• Keep going until every factor is prime
  → Writing a prime factorisation
• Add exponents when pieces share a prime
  → Writing a prime factorisation
• Distinct is not counted with repeats
  → Reading a factorisation: distinct primes, repeats and the number itself
• A bare exponent counts once
  → Reading a factorisation: distinct primes, repeats and the number itself
• Add 1 to each exponent first
  → How many divisors a number has
• Factorise fully before counting
  → How many divisors a number has
• Even divisors are not half the total
  → Odd divisors, even divisors and the sum of the divisors

Formulas (4)

• Factors, multiples and the common ones · Factor pairs
  $$n = a \times b \;\Rightarrow\; a \text{ and } b \text{ are both factors of } n$$
• HCF, by primes or by Euclid, and co-prime numbers · HCF from prime factors
  $$\text{HCF} = \text{product of the common primes, each to its lowest power}$$
• LCM, and the least number leaving a given remainder · LCM from prime factors
  $$\text{LCM} = \text{product of every prime present, each to its highest power}$$
• HCF × LCM = product of the two numbers · Two numbers
  $$\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$$

Watch out for (7)

• Factors versus multiples
  → Factors, multiples and the common ones
• Do not drop 1 or the number itself
  → Factors, multiples and the common ones
• Co-prime is not prime
  → HCF, by primes or by Euclid, and co-prime numbers
• Lowest powers for HCF
  → HCF, by primes or by Euclid, and co-prime numbers
• The LCM is not always the product
  → LCM, and the least number leaving a given remainder
• Add the remainder, do not subtract it
  → LCM, and the least number leaving a given remainder
• Two numbers only
  → HCF × LCM = product of the two numbers

Formulas (3)

• Dividend = divisor × quotient + remainder · Division algorithm
  $$N = d \times q + r,\qquad 0 \le r < d$$
• Remainders of sums, multiples and products · Working with remainders
  $$a \equiv r_a,\ b \equiv r_b \pmod d \;\Rightarrow\; a + b \equiv r_a + r_b,\quad ab \equiv r_a r_b \pmod d$$
• Remainders of large powers · Splitting the exponent
  $$a^k \equiv 1 \pmod d \;\Rightarrow\; a^{kq + s} \equiv a^s \pmod d$$

Watch out for (6)

• The remainder is smaller than the divisor
  → Dividend = divisor × quotient + remainder
• Multiply back to check
  → Dividend = divisor × quotient + remainder
• Reduce the answer
  → Remainders of sums, multiples and products
• The digit-sum shortcut is for 3 and 9 only
  → Remainders of sums, multiples and products
• Minus 1 means one less than the divisor
  → Remainders of large powers
• Use the full cycle length
  → Remainders of large powers