Unit 1

Test of Divisibility

If a number is given and we want to know which number can divide it, there are some rules:

• 2: All even numbers are divisible by 2.
• 3: If the sum of digits is divisible by 3, then that number will be divisible by 3. For example, in 4331, 4+3+3+1 = 11, which is not divisible by 3, so 4331 is not divisible by 3. On the other hand, for 4332, 4+3+3+2 = 12, which is divisible by 3, so 4332 is also divisible by 3.
• 4: Check the last 2 digits. If the last 2 digits are divisible by 4, then the whole number will be divisible by 4. For example, in 136, 36 is divisible by 4, so 136 is also divisible by 4.
• 5: Ends in 0 or 5.
• 6: If a number is divisible by both 2 and 3, then it will be divisible by 6 as well.
• 7: Double the last digit and subtract it from the remaining digits. If the result is divisible by 7, then the whole number is divisible by 7. For example, in 427, double the last digit 7 (7 * 2 = 14) and subtract it from the remaining digits 42 (42 - 14 = 28), which is divisible by 7, so 427 is divisible by 7.
• 8: Check the last 3 digits. If the last 3 digits are divisible by 8, then the whole number will be divisible by 8. For example, in 1064, 064 is divisible by 8, so 1064 is also divisible by 8.
• 9: The sum of digits should be a multiple of 9. For example, in 5283, 5+2+8+3 = 18, which is divisible by 9, so 5283 is also divisible by 9.
• 11: If the sum of the odd digits minus the sum of the even digits is a multiple of 11 or 0, then the number is divisible by 11. For example, in 121, 1+1 - 2 = 0, hence it is divisible by 11.
• 12: If a number is divisible by both 3 and 4, then it will be divisible by 12.

Factors and Multiples

• If a number, say, 'a' exactly divides another number 'b', we can say that 'a' is a factor of 'b'. Further, 'b' is called a multiple of 'a'.

Total factor, even factor, odd factor and prime factors

Another example:

Concept of Unit Digit

Q- what will be the unit digit of the following:
34¹⁹⁶ * 39²³⁵ * 60¹²³?

LCM and HCF

• LCM: The LCM of two or more numbers is the smallest multiple that is divisible by each of the numbers without leaving a remainder. In other words, it is the smallest common multiple of the given numbers.
  For example, the LCM of 4 and 6 is 12 because 12 is the smallest multiple that both 4 and 6 share without leaving any remainder when divided by either of them.
• The HCF or GCD of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder. In other words, it is the greatest common divisor of the given numbers.
  For example, the HCF of 12 and 18 is 6 because 6 is the largest number that both 12 and 18 can be divided evenly.

HCF and LCM of Fractions

Q- The HCF of two numbers is 5, and the LCM is 30. If one of the two numbers is 15, find the sum of those two numbers.

LCM and HCF remainder Questions

Categories of question in LCM and HCF.

Cases in LCM type quesitons

If the question asks to find the largest 3-digit number, then K will be the number that, when multiplied, gives the largest 3-digit number. For example, if the LCM is 60, the largest 3-digit number will be 60K, where K is the largest integer that makes the product a 3-digit number. In this case, K would be 16 because 60 × 16 = 960, which is the largest 3-digit number that is a multiple of 60.

Cases in HCF type quesitons

Questions:

1. Find smallest number divisible by 2, 4 and 6.
2. Which greatest number will divide 3026 and 5053 leaving remainders 11 and 13 respectively.
   1. 15
   2. 30
   3. 45
   4. 60
3. The greatest number, which when divides 989 and 1327 leave remainders 5 and 7 respectively.
   1. 8
   2. 16
   3. 24
   4. 32
4. The least number, which when divided by 12, 15, 20 or 54 leaves a remainder of 4 in each case is.
   1. 450
   2. 454
   3. 540
   4. 544
5. The smallest number, which is divided by 18, 27 and 36 separately leaves remainder 5, 14 and 23 respectively, is-
   1. 95
   2. 113
   3. 149
   4. 77
6. What is the smallest number, which when is subtracted from 1936 and resulting number is divided by any of as 9, 10 and 15 leaves remainder 7?
   1. 37
   2. 36
   3. 39
   4. 30
7. The largest number by which when 61, 72 and 83 is divided, the remainder is 1, 2 and 3 respectively.
   1. 20
   2. 15
   3. 12
   4. 10
8. The largest number of five digits which, when divided by 16, 24, 30 or 36 leaves the same remainder 10 in each case is.
   1. 99279
   2. 99370
   3. 99269
   4. 99350
9. Find the greatest number of five digits which when divided by 3, 5, 8, 12 leaves 2 as remainder.
   1. 99999
   2. 99958
   3. 99960
   4. 99962
10. The least multiple of 13 which on dividing by 4, 5, 6, 7 and 8 leaves remainder 2 in each case.
    1. 2520
    2. 842
    3. 2522
    4. 840

Answers: 2-c, 3-c, 4-d, 5-a, 6-c, 7-d, 8-b, 9-d, 10-c.

More questions:

1. What should be added or subtracted from the number 713 so that the new number is divisible by 2, 3, and 7?
2. Find the largest 3-digit number that will give a remainder of 4 when divided by 5, 6, and 8.
3. How many numbers from 1 to 200 will give a remainder of 3 when divided by 4 and 6?
4. What is the smallest number that gives a remainder of 3 when divided by 7 and a remainder of 5 when divided by 9?
5. Find the number greater than 500 that gives remainders of 2, 3, and 4 when divided by 5, 6, and 7, respectively.
6. Find the smallest 3-digit number that gives remainders of 5 and 3 when divided by 6 and 5, respectively.