Tips on cracking Aptitude Questions on Number System

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5 TIPS on cracking Aptitude Questions on Number
Systems
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Tip 1 Factorize the expression

• (a b)(a - b) (a2 - b2)
• (a b)2 (a2 b2 2ab)
• (a - b)2 (a2 b2 - 2ab)
• (a b c)2 a2 b2 c2 2(ab bc ca)
• (a3 b3) (a b)(a2 - ab b2)
• (a3 - b3) (a - b)(a2 ab b2)
• (a3 b3 c3 - 3abc) (a b c)(a2 b2 c2
  - ab - bc ac)
• Question 297 x 297 ?
• Solution
• 297 x 297 (300 3)(300 3) (300 3)2
  3002 32 2x3x300
• 297 x 297 90000 9 1800 88209
• Question 753 x 753 247 x 247 - 753 x 247 ?
• 753 x 753 x 753 247 x
  247 x 247
• Solution

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Tip 2 Use divisibility tests where applicable

• Question How many 3-digit numbers are completely
  divisible 6 ?
• Solution
• Smallest 3-digit number divisible by 6
• 100 is not divisible by 3.
• 101 is not divisible by 2.
• 102 is divisible by both 2 and 3 and hence, by 6.
• Largest 3-digit number divisible by 6
• 999 is not divisible by 2.
• 998 is not divisible by 3.
• 997 is not divisible by 2.
• 996 is divisible by both 2 and 3 and hence,
  divisible by 6.
• Thus, the numbers form an AP with a 102, d 6,
  l 996, n ?
• 102 (n 1)6 996
• 6n 996 102 6 900
• n 900 / 6 150
• Question If 481d673 is divisible by 9, what
  should be the smallest whole number in place of d
  ?

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Tip 3 Apply the rules of Binomial Theorem

• n
• (x y)n S nCrx(n r)yr
• r0
• (1 x)-1 1 x x2 x3 . where x is
  between 0 and 1
• (1 x) -1 1 x x2 x3 where x is
  between 0 and 1
• xn 1 is divisible by x 1 for all odd
  values of n.
• Proof Let us prove this formula using the method
  of induction.
• When n1, xn 1 x1 which is divisible by
  (x1). When n2, x2 1 is not divisible by
  (x1).
• Let x2k1 1 be divisible by (x1).
• When n 2k3 (the next odd number after 2k 1)
• x2k3 1 x2(x2k1 1) (1 - x2) x2(x2k1
  1)(1x)(1-x), which is divisible by (x1).
• Thus, xn 1 is divisible by x 1 when n is odd.
  
• xn 1 is divisible by x 1 only when n is
  even. Proof is similar to the one above
• Question What will be remainder when (6767 67)
  is divided by 68 ?
• Solution
• From Binomial Theorem, we know, xn 1 is
  divisible by x 1 only when n is odd.

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Tip 4 Remember the rules for division

• Dividend (Divisor x Quotient) Remainder
• Any recurring decimal can be written as the sum
  of a non-recurring decimal and a fraction of the
  recurring portion of the decimal out of (10n
  1).
• Ex 0.125125125 0.125 125/999 and
  7.2341341341 7.2 341/9999
• Question Express 0.232323..... As a rational
  number.
• Solution
• 0.232323 23/99.
• Question A student mistook the divisor as 12
  instead of 21 and obtained 35 as quotient and
  reminder as 0. What is the correct quotient ?
• Solution
• Let the dividend be x.
• Then, x / 12 35
• x 12 x 35
• Correct quotient (12 x 35 / 21) 4 x 5 20.
• Question Find the smallest 6 digit number
  exactly divisible by 111.

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Tip 5 Memorize the formulae for sum of powers
and apply where suitable

• 1 2 3 .. n n (n 1) / 2
• 12 22 32 .. n2 n (n 1)(2n 1) / 6
• 13 23 33 ... n3 n (n 1) / 2 2
• Question  (51 52 53 ... 100) ?
• Solution
• (51 52 53 .. 100) (1 2 3 ..
  100) (1 2 3 .. 50)
• 100(100 1) - 50(50 1)
• 2 2
• 5050 1275
• 3775.
• Question (22  42  62  ... 202) ?
• Solution (22  42  62  ... 202) 22(12
  22 32 .. 102)
• 22 x 10(10 1)(210 1)
• 6
• 4 x 10 x 11 x 21 / 6
• 1540.

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