Fermat Theorem

Q.

Three numbers which are co-prime to each other are such that the product of the first two is 551 and that of the last two is 1073. The sum of the three numbers is :

1. 93

2. 75

3. 81

4. 85

5. 89

Q.

List the first five multiples of $20$.

Q.

At a conference, every member attending the event makes a handshake with every other member. If there were 105 handshakes in all, what was the number of members attending the conference?

1. 15

2. 14

3. 13

4. 16

5. 20

Q.

List the first five multiples of $16$.

Q.

The sides of a pentagonal field (not regular) are $1737\mathrm{m},2160\mathrm{m},2358\mathrm{m},1422\mathrm{m}\mathrm{and}2214\mathrm{m}\mathrm{respectively}.$ Find the greatest length of the tape by which the five sides may be measured completely.

1. $7$

2. $13$

3. $11$

4. $9$

Q.

The common factor of 3ab and 2cd is

1. $1$

2. $-1$

3. $a$

4. $c$

Q. When $2^{256}$ is divided by 17, the remainder would be : (CAT 2002)

1. 1
2. 16
3. 14
4. None of these

Q. Directions: Study the following table carefully and answer the questions that follows:
Chart showing schedule of train from Dadar to Bhuj and number of passengers boarding from each station

Distance between which two stations is second lowest?

1. Nadiad Jn. to Ahmedabad
2. Dadar to Vasai Road
3. Anand Jn. to Vadodara
4. Anand Jn. to Nadiad Jn
5. Vasai Road to Surat

Q. The remainder, when $({15}^{23}+{23}^{23})$ is divided by 19, is : (CAT 2004)

1. 18
2. 4
3. 15
4. 0

Q.

The remainder obtained when ${23}^{21}-{19}^{21}$ is divisible by 21 is

1. 0

2. 1

3. 20

4. 16

Q.

Mention at least ten pairs of co-prime numbers between $1$ and $100$ .

Q. Given that p is a prime number. $p^2[p^{27}-\frac{1}{p}]$ is always divisible by?

1. 23
2. 27
3. none of these
4. 25

Q. Find the remainder when ${10}^{1000}$ is divided by 7.

1. 2
2. 4
3. 3
4. –1

Q.

When a person walks at $\frac{4}{5}$th of his usual speed, he is late by 5 hrs. The usual time is

1. 17 hrs

2. 16 hrs

3. 20 hrs

4. 25 hrs

Q. How many natural numbers between 1 and 900 are NOT multiples of any of the numbers 2, 3, or 5?

1. 240
2. 250
3. 270
4. 210
5. 300

Q. Find the sum of all the co-primes to 2016 which are less than 2016

1. $552108$
2. $5016$
3. $558432$
4. $580608$

Q. Find the remainder, if $2^4$ $\times 27$ divided by $6$.

1. 1
2. 3
3. 2
4. 0

Q.

Let $A=\left\{\right(2,5,11\left)\right\},B=\{3,6,10$} and $R$ be a relation from $A$ to $B$ defined by $R=(a,b):aandbareco-prime$. Then $R$ is

1. $(2,3),(2,6),(5,10),(5,6)$

2. $(2,6),(2,10),(5,10)$

3. $(2,3),(5,3),(5,6),(11,3),(11,6),(11,10)$

4. $(2,10),(5,3),(5,6),(11,10).$