Number System
20 Questions
Practice Problems
Number System Interview Questions
20 Questions
Number System — Concept Notes
A quick refresher on the core formulas for each operation. Read through this once, then head into practice.
- Sum of first n natural numbers: n(n+1) / 2
- Sum of first n even numbers: n(n+1)
- Sum of first n odd numbers: n²
- Place value + face value: every digit's place value is the digit multiplied by its position's power of 10; adding these across all digits reconstructs the number.
- The above method is best illustrated with the help of following example:
Example
Illustration 2: 54321 − (9876 + 8967 + 7689) = ? Step 1 — Add 1st column: 6 + 7 + 9 = 22 To obtain 1 at unit's place add 9 to make 31. In the answer, write 9 at unit's place and carry over 3. Step 2 — Add 2nd column: 3 + 7 + 6 + 8 = 24 To obtain 2 at ten's place, add 8 to make 32. In the answer, write 8 at ten's place and carry over 3. Step 3 — Add 3rd column: 3 + 8 + 9 + 6 = 26 To obtain 3 at hundred's place, add 7 to make 33. In the answer, write 7 at hundred's place and carry over 3. Step 4 — Add 4th column: 3 + 9 + 8 + 7 = 27 To obtain 4 at thousand's place, add 7 to make 34. In the answer, write 7 at thousand's place and carry over 3. Step 5 — 5th column: To obtain 5 at ten-thousand's place add 2 to it to make 5. In the answer, write 2 at the ten-thousand's place. ∴ 54321 − (9876 + 8967 + 7689) = 27789.
- 1. Multiplication of a given number by 9, 99, 999, etc., that is by 10n−1 Method: Put as many zeros to the right of the multiplicant as there are nines in the multiplier and from the result subtract the multiplicant and get the answer.
- 2. Multiplication of a given number by 11, 101, 1001, etc., that is by 10n+1 Method: Place n zeros to the right of the multiplicant and then add the multiplicant to the number so obtained.
- Distributive Laws For any three numbers a,b,c, we have: (a) a×b+a×c=a×(b+c) (b) a×b−a×c=a×(b−c)
Example
Illustration : Multiply: (a) 3893 by 99 (b) 4327 by 999 (c) 5863 by 9999 Solution: (a) 3893 × 99 = 389300 − 3893 = 385407 (b) 4327 × 999 = 4327000 − 4327 = 4322673 (c) 5863 × 9999 = 58630000 − 5863 = 58624137 Illustration : Multiply: (a) 4782 × 11 (b) 9836 × 101 (c) 6538 × 1001 Solution: (a) 4782 × 11 = 47820 + 4782 = 52602 (b) 9836 × 101 = 983600 + 9836 = 993436 (c) 6538 × 1001 = 6538000 + 6538 = 6544538
- Division is repeated subtraction.
- For example, when we divide 63289 by 43, it means 43 can be repeatedly subtracted 1471 times from 63289 and the remainder 36 is left.
- Dividend=(Divisor×Quotient)+Remainder
- or,
- Divisor = (Dividend − Remainder) / Quotient
Example
Illustration : On dividing 7865321 by a certain number, the quotient is 33612 and the remainder is 113. Find the divisor. Solution: Divisor = (Dividend − Remainder) / Quotient = (7865321 − 113) / 33612 = 7865208 / 33612 = 234 Illustration : A number when divided by 315 leaves remainder 46 and the value of quotient is 7. Find the number. Solution: Number=(Divisor×Quotient)+Remainder=(315×7)+46=2205+46=2251
- 1. Divisibility by 2 A number is divisible by 2 if the unit's digit is zero or divisible by 2. For example, 4, 12, 30, 18, 102, etc., are all divisible by 2.
- 2. Divisibility by 3 A number is divisible by 3 if the sum of digits in the number is divisible by 3. For example, the number 3792 is divisible by 3 since: 3+7+9+2=21 which is divisible by 3.
- 3. Divisibility by 4 A number is divisible by 4 if the number formed by the last two digits (ten's digit and unit's digit) is divisible by 4 or both are zero. For example, the number 2616 is divisible by 4 since 16 is divisible by 4.
- 4. Divisibility by 5 A number is divisible by 5 if the unit's digit in the number is 0 or 5. For example, 13520, 7805, 640, 745, etc., are all divisible by 5.
- 5. Divisibility by 6 A number is divisible by 6 if the number is even and sum of its digits is divisible by 3. For example, the number 4518 is divisible by 6 since it is even and the sum of its digits: 4+5+1+8=18 is divisible by 3.
- 6. Divisibility by 7 The unit digit of the given number is doubled and then it is subtracted from the number obtained after omitting the unit digit. If the remainder is divisible by 7, then the given number is also divisible by 7. For example, consider the number 448. On doubling the unit digit 8 of 448 we get 16. Then, 44−16=28 Since 28 is divisible by 7, 448 is divisible by 7.
- 7. Divisibility by 8 A number is divisible by 8, if the number formed by the last 3 digits is divisible by 8. For example, the number 41784 is divisible by 8 as the number formed by last three digits, i.e., 784 is divisible by 8.
- 8. Divisibility by 9 A number is divisible by 9 if the sum of its digits is divisible by 9. For example, the number 19044 is divisible by 9 as the sum of its digits: 1+9+0+4+4=18 is divisible by 9.
- 9. Divisibility by 10 A number is divisible by 10, if it ends in zero. For example, the last digit of 580 is zero, therefore, 580 is divisible by 10.
- 10. Divisibility by 11 A number is divisible by 11 if the difference of the sum of the digits at odd places and sum of the digits at even places is either zero or divisible by 11. For example, in the number 38797, the sum of the digits at odd places is: 3+7+7=17 and the sum of the digits at even places is: 8+9=17 The difference is: 17−17=0 so, the number is divisible by 11.
- 11. Divisibility by 12 A number is divisible by 12 if it is divisible by 3 and 4.
- 12. Divisibility by 18 An even number satisfying the divisibility test of 9 is divisible by 18.
- 13. Divisibility by 25 A number is divisible by 25 if the number formed by the last two digits is divisible by 25 or the last two digits are zero. For example, the number 13675 is divisible by 25 as the number formed by the last two digits is 75, which is divisible by 25.
- 14. Divisibility by 88 A number is divisible by 88 if it is divisible by 11 and 8.
- 15. Divisibility by 125 A number is divisible by 125 if the number formed by the last three digits is divisible by 125 or the last three digits are zero. For example, the number 5250 is divisible by 125 as 250 is divisible by 125.