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
The remainder would be given by: (5 + 7 + 10 + 23 + 27)/34 = 72/34 remainder = 4.
Ans .
22
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
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
43197 / 7 is same as 17/7 = 1
Ans .
51203 / 7 = 2203 / 7 = (23)67 * 22 / 7 = 167 * 4 / 7 = 4
Ans .
4
5928 / 7 = 328 / 7 = (32)14 / 7 = (2)14 / 7 = 1024 * 16 / 7 = 2 * 2 / 7 = 4
Ans .
4177 / 7 = (-1)77 / 7 = 6
Ans .
4
25102 / 17 = 8102 / 17 = (23)102 / 17 = 2306 / 17 = (24)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 1468 .
In such a case, follow the following structure for conversion: 1468 = 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 63 = 216 is contained in 347, while the value of 64 = 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 63, 62, 61, 60 respectively. Hence, a 4-digit number. Structure of number: - - - -
Step 2 : Now, find out how many times can 63 be accommodated in 347. It can be accommodated only once as 2 * 63 = 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 62 can be accommodated in 131,answer is 3 times. As 3 * 62 = 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 61 be accommodated in 23, The answer is 3 times. 6 * 61 = 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 60 be accommodated in 5, The answer is 5 times. 5 * 60 = 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 56 = 57 = 58 = 59 = 510
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 nx – n(x - 1). Example : The number of 2 digit numbers in binary system is 22 – 2(2 - 1) = 2
Ans .
1100
The number is a four digit number as 23 can be accomodated in it. So we start with 23 - 20 . Using above logic we get 1100
Ans .
1192
1 × 123 + 1 × 122 + 9 × 121 + 2 × 120 = 1728 + 144 + 108 + 2 = 1982. Hence number is 1192
Ans .
65
(203)5 = (?)10 = 2 × 52 + 0 × 51 + 3 × 50 = 50 + 0 + 3 = 5310. This is 65 in base 8.
Ans .
75
(52)7 = (5 × 71 + 2 × 70)10 = (37)10 also, (46)8 = (4 × 81 + 6 × 80)10 = (38)10 sum = (75)10
The units place digit is checked and then its power is checked. If the number is 3222 then its same as 34n+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 1722 will have same units digit as 72 and 77722 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
37123 = 74n+3 = 3
Ans .
43144 = 34n = 1
Ans .
8
(173)45 × (152)77 × (777)999 = (3)4n+1 * (2)4n+1 * (7)4n+3 = 3*2*3 = 8