1421 * 1423 * 1425 / 12 same as R (1421/12) * R(1423/12) * R(1425/12) where R(p/q) is remainder of p/q.

Therefore we get 5 * 7 * 9 / 12 = 35 * 9 / 12 = 99 / 12 = 3.

11 * 10 * 9 / 12 same as NR (11/12) * NR(10/12) * NR(9/12) where NR(p/q) is negative remainder of p/q.

So if 35/12 then remainder shall be 11 and negative remainder shall 11 - 12 = -1

(ax + 1)

37

35

1421 * 1423 * 1425 / 100 same as R (1421/100) * R(1423/100) * R(1425/100) where R(p/q) is remainder of p/q.

Therefore we get 21 * 23 * 25 / 100 = 525 * 23 / 100 ; 25 * 23 / 100 = 575 / 100

Thus the last two digits are 75.

1421 * 1423 * 1425 / 10 same as R (1421/10) * R(1423/10) * R(1425/10) where R(p/q) is remainder of p/q.

Therefore we get 1 * 3 * 5 / 10 = 15 / 10 = 5

Thus the last digit is 5.

If you can express the expression in the form \( \frac{(ax + 1)^n}{a} \) the remainder will become 1 directly. In such a case, no matter how large the value of the power n is, the remainder is 1.

**Example :**\( \frac{(37^{12654})}{9} = \frac{(9*4 + 1)^{12654}}{9} = 1\)If you can express the expression in the form \( \frac{(ax - 1)^n}{a} \) the remainder will become 1 directly. In such a case, no matter how large the value of the power n is, the remainder is 1.

**Example :**\( \frac{(35^{12654})}{9} = \frac{(9*4 - 1)^{12654}}{9} = -1\). The remainder cannot be negative and so -1 remainder means 8.If you can express the expression in the form \( \frac{(a)^n}{a+1} \) the remainder will become "a" directly if "n" is odd and 1 if "n" is even.

If you can express the expression in the form \( \frac{(a+1)^n}{a} \) the remainder will become 1 for any power "n".

**Ans . **

4

**Explanation :**The remainder would be given by: (5 + 7 + 10 + 23 + 27)/34 = 72/34 remainder = 4.

**Ans . **

22

**Explanation :**The remainder would be given by: (5 * 7 * 10 * 23 * 27)/34 = 35 * 230 * 27/ 34 = 1 * 26 * 27/34 = 702/34 So remainder = 22

**Ans . **

32

**Explanation :**The remainder would be given by: (5 * 7 * 10 * 23 * 27 * 3)/34 = 35 * 230 * 27 * 3/34 = 1 * 26 * 81/34 = 26 * 13/34 = 338/34 So remainder = 32

**Ans . **

1

**Explanation :**43

^{197}/ 7 is same as 1^{7}/7 = 1

**Ans . **

**Explanation :**51

^{203}/ 7 = 2^{203}/ 7 = (2^{3})^{67}* 2^{2}/ 7 = 1^{67}* 4 / 7 = 4

**Ans . **

4

**Explanation :**59

^{28}/ 7 = 3^{28}/ 7 = (3^{2})^{14}/ 7 = (2)^{14}/ 7 = 1024 * 16 / 7 = 2 * 2 / 7 = 4

**Ans . **

**Explanation :**41

^{77 / 7 = (-1)77 / 7 = 6}

**Ans . **

4

**Explanation :**25

^{102}/ 17 = 8^{102}/ 17 = (2^{3})^{102}/ 17 = 2^{306}/ 17 = (2^{4})^{76 * 22 / 17 = (-1)76 * 4 / 17 = 4/17 = Remainder is 4}

All the work we carry out in our number system is called as the decimal system. In other words we work in the decimal system.

**Why is it called decimal?? It is because there are 10 digits in the system 0–9.**However, depending on the number of digits contained in the base system other number systems are also possible. Thus a number system with base 2 is called the binary number system and will have only two digits 0 and 1. Some of the most commonly used systems are: Binary (base 2), Octal (base 8), Hexadecimal (base 16).

Binary system has 2 digits : 0, 1. Octal has 8 digits : – 0, 1, 2, 3, ... 7.

Hexadecimal has 16 digits – 0, 1, 2, ... 9, A, B, C, D, E, F; Where A has a value 10, B = 11 and so on

**Conversion from any base system into decimal:**

Suppose you have to write the decimal equivalent of the base 8 number 146

_{8}.In such a case, follow the following structure for conversion: 146

_{8}= 1 × 8^{ 2}+ 4 × 8^{ 1}+ 6 × 8^{ 0}= 64 + 32 + 6 = 102

** Conversion of a number in decimals into any base:**

Suppose you have to find out the value of the decimal number 347 in base 6. The following process is to be adopted:

**Step 1 :**Find the highest power of the base (6 in this case) that is contained in 347. In this case you will realise that the value of 6^{3}= 216 is contained in 347, while the value of 6^{4}= 1296 is not contained in 347. Hence, we realise that the highest power of 6 contained in 347 is 3. This should make you realize that the number has to be constructed by using the powers 6^{3}, 6^{2}, 6^{1}, 6^{0}respectively. Hence, a 4-digit number.**Structure of number: - - - -****Step 2 :**Now, find out how many times can 6^{3}be accommodated in 347. It can be accommodated only once as 2 * 6^{3}= 432 , which is greater than original number. So the first digit is 1 and the number becomes 1 _ _ _.**Step 3 :**Now 347 - 216 = 131, We check how many times 6^{2}can be accommodated in 131,answer is 3 times. As 3 * 6^{2}= 108, So the second digit is 3 and the number becomes 1 3 _ _. Remainder term = 131 - 108 = 23.**Step 4 :**Now we check how many times can 6^{1}be accommodated in 23, The answer is 3 times. 6 * 6^{1}= 18. So the third digit is 3 and the number becomes 1 3 3 _. Remainder term = 23 - 18 = 5.**Step 5 :**Now we check how many times can 6^{0}be accommodated in 5, The answer is 5 times. 5 * 6^{0}= 5. Thus we get the number as 1335.

In single digits there is no difference between the value of the number—whichever base we take. Illustrations For example, the equality 5

_{6}= 5_{7}= 5_{8}= 5_{9}= 5_{10}Suppose you have a number in base x. When you convert this number into its decimal value, the value should be such that when it is divided by x, the remainder should be equal to the units digit of the number in base x.

The number of x digit numbers in nth base system will be n

^{x}– n^{(x - 1)}. Example : The number of 2 digit numbers in binary system is 2^{2}– 2^{(2 - 1)}= 2

**Ans . **

1100

**Explanation :**The number is a four digit number as 2

^{3}can be accomodated in it. So we start with 2^{3}- 2^{0}. Using above logic we get 1100

**Ans . **

1192

**Explanation :**1 × 12

^{3}+ 1 × 12^{2}+ 9 × 12^{1}+ 2 × 12^{0}= 1728 + 144 + 108 + 2 = 1982. Hence number is 1192

**Ans . **

65

**Explanation :**(203)

_{5}= (?)_{10}= 2 × 5^{2}+ 0 × 5^{1}+ 3 × 5^{0}= 50 + 0 + 3 = 53_{10}. This is 65 in base 8.

**Ans . **

75

**Explanation :**(52)

_{7}= (5 × 7^{1}+ 2 × 7^{0})10 = (37)10 also, (46)_{8}= (4 × 8^{1}+ 6 × 8^{0})_{10}= (38)_{10}sum = (75)_{10}

The units place digit is checked and then its power is checked. If the number is 3

^{222}then its same as 3^{4n+2 = 9. 3222 will also have remainder as 9}Since for numbers with many digits we only check the unit digit raised to the given power. We say 17

^{22}will have same units digit as 7^{2}and 777^{22}etc.

Number ending in | If the value of the Power is | |||
---|---|---|---|---|

4n + 1 | 4n + 2 | 4n + 3 | 4n + 4 | |

1 | 1 | 1 | 1 | 1 |

1 | 2 | 4 | 8 | 6 |

3 | 3 | 9 | 7 | 1 |

4 | 4 | 6 | 4 | 6 |

5 | 5 | 5 | 5 | 5 |

6 | 6 | 6 | 6 | 6 |

7 | 7 | 9 | 3 | 1 |

8 | 8 | 4 | 2 | 6 |

9 | 9 | 1 | 9 | 1 |

**Ans . **

3

**Explanation :**37

^{123}= 7^{4n+3}= 3

**Ans . **

**Explanation :**43

^{144}= 3^{4n}= 1

**Ans . **

8

**Explanation :**(173)

^{45}× (152)^{77}× (777)^{999}= (3)^{4n+1}* (2)^{4n+1}* (7)^{4n+3}= 3*2*3 = 8