Number System: SSC CGL Quantitative Aptitude

Formulas (2)

• When a fraction has a terminating decimal · Terminating decimal test
  $$\frac{p}{q} \ (\text{lowest terms}) \text{ terminates exactly when } q = 2^a \times 5^b$$
• Perfect squares near a given number · Bracketing between squares
  $$k^2 \le N < (k + 1)^2: \ \text{add } (k + 1)^2 - N \text{ or subtract } N - k^2 \qquad (a + b)^2 = a^2 + 2ab + b^2$$

Reference tables (1)

Watch out for (8)

• Calling 0 a natural number
  → The families of numbers and how they nest
• Saying rational times irrational is always irrational
  → The families of numbers and how they nest
• Taking a repeating decimal for an irrational number
  → The families of numbers and how they nest
• Judging the denominator before cancelling
  → When a fraction has a terminating decimal
• Counting 1 as a prime
  → Prime numbers and the square-root test
• Letting 51, 57 or 91 through as prime
  → Prime numbers and the square-root test
• Going to the wrong square
  → Perfect squares near a given number
• Trusting the last digit as proof
  → Perfect squares near a given number

Formulas (2)

• The division algorithm · Division algorithm
  $$\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}, \qquad 0 \le \text{Remainder} < \text{Divisor}$$
• The nearest multiple of a number · Distance to the nearest multiples
  $$N = dq + r: \quad \text{subtract } r, \ \text{or add } d - r$$

Watch out for (4)

• Accepting a remainder as large as the divisor
  → The division algorithm
• Scaling the quotient of a division that had a remainder
  → The division algorithm
• Adding the remainder instead of the gap
  → The nearest multiple of a number
• Landing on a multiple with the wrong number of digits
  → The nearest multiple of a number

Formulas (3)

• The alternating-sum test for 11 · Test for 11
  $$D = (\text{sum of digits in odd places}) - (\text{sum of digits in even places}), \quad 11 \text{ divides } N \text{ exactly when } D = 0, \pm 11, \pm 22, \ldots \qquad \overline{abcabc} = \overline{abc} \times 7 \times 11 \times 13$$
• Testing a composite divisor by its co-prime parts · Common co-prime splits
  $$6 = 2 \times 3 \quad 12 = 3 \times 4 \quad 18 = 2 \times 9 \quad 24 = 3 \times 8 \quad 33 = 3 \times 11 \quad 36 = 4 \times 9 \quad 44 = 4 \times 11 \quad 45 = 5 \times 9 \quad 72 = 8 \times 9 \quad 99 = 9 \times 11$$
• Remainders from the digit tests · Remainders from digits
  $$N \div 9 \text{ leaves the remainder of (digit sum)} \div 9 \qquad N \div 11 \text{ leaves the remainder of } D \div 11$$

Reference tables (1)

Watch out for (8)

• Testing 8 on the last two digits
  → Divisibility tests for 2, 3, 4, 5, 7, 8, 9, 10, 11 and 25
• Treating divisible by 3 as divisible by 9
  → Divisibility tests for 2, 3, 4, 5, 7, 8, 9, 10, 11 and 25
• Rejecting a difference of 11
  → The alternating-sum test for 11
• Skipping a zero digit
  → The alternating-sum test for 11
• Splitting a divisor into parts that share a factor
  → Testing a composite divisor by its co-prime parts
• Running only one of the two tests
  → Testing a composite divisor by its co-prime parts
• Reading the 11 difference from the left
  → Remainders from the digit tests
• Giving a negative difference as the remainder
  → Remainders from the digit tests

Formulas (2)

• A missing digit from a digit-sum test · Hidden digit from a digit sum
  $$S + x = 9k \ (\text{or } 3k), \quad 0 \le x \le 9$$
• A missing digit from a last-digits test · Last-digit tests for 8 and 4
  $$\overline{abc} \text{ is a multiple of } 8: \ a \text{ even} \Rightarrow 8 \text{ divides } \overline{bc}; \ a \text{ odd} \Rightarrow 8 \text{ divides } \overline{bc} + 4 \qquad 4 \text{ divides } \overline{bc} \text{ exactly when } 4 \text{ divides } 2b + c$$

Watch out for (5)

• Accepting a value that is not a digit
  → A missing digit from a digit-sum test
• Missing the second answer for 9
  → A missing digit from a digit-sum test
• Letting a multiple of 8 and 5 end in 5
  → A missing digit from a last-digits test
• Running the 8 test on two digits
  → A missing digit from a last-digits test
• Setting the difference to 0 and stopping
  → A missing digit from the test for 11

Formulas (3)

• Prime factorisation and counting divisors · Divisors from the prime factorisation
  $$N = p^a q^b r^c: \quad \text{number of divisors} = (a + 1)(b + 1)(c + 1) \qquad \text{sum of divisors} = (1 + p + \cdots + p^a)(1 + q + \cdots + q^b)(1 + r + \cdots + r^c) \qquad 1 + 2 + \cdots + n = \frac{n(n + 1)}{2}$$
• Taking a common power out of a sum of powers · Sums and differences of powers
  $$a^m + a^{m+1} + \cdots + a^{m+k} = a^m(1 + a + \cdots + a^k) \qquad a^n - b^n \text{ is divisible by } a - b \qquad a^n - b^n \text{ is divisible by } a + b \ (n \text{ even}) \qquad a^n + b^n \text{ is divisible by } a + b \ (n \text{ odd})$$
• HCF, LCM and the least square multiple · HCF, LCM, tiles and the mean proportional
  $$\text{HCF} \times \text{LCM} = a \times b \qquad \sqrt{ab} = \sqrt{\text{HCF} \times \text{LCM}} \qquad \text{tiles} = \frac{L}{h} \times \frac{B}{h}, \ h = \text{HCF}(L, B)$$

Watch out for (5)

• Multiplying the powers instead of the powers plus one
  → Prime factorisation and counting divisors
• Forgetting the factors of the power you took out
  → Taking a common power out of a sum of powers
• Using a + b on an odd power of a difference
  → Taking a common power out of a sum of powers
• Using HCF times LCM for three numbers
  → HCF, LCM and the least square multiple
• Squaring the LCM to get a square multiple
  → HCF, LCM and the least square multiple

Formulas (2)

• Replacing a number by its remainder · Swapping numbers for remainders
  $$N = dk + r \Rightarrow N \equiv r \pmod d \qquad a \equiv r, \ b \equiv s \pmod d \Rightarrow a + b \equiv r + s, \ ab \equiv rs \pmod d$$
• Remainders of large powers · Powers when the base leaves 1 or one less than the divisor
  $$a \equiv 1 \pmod d \Rightarrow a^n \equiv 1 \qquad a \equiv -1 \pmod d \Rightarrow a^n \equiv 1 \ (n \text{ even}), \ a^n \equiv d - 1 \ (n \text{ odd})$$

Watch out for (4)

• Moving to a divisor that does not divide the first one
  → Replacing a number by its remainder
• Leaving the answer bigger than the divisor
  → Replacing a number by its remainder
• Writing minus one as the remainder
  → Remainders of large powers
• Ignoring whether the exponent is odd or even
  → Remainders of large powers