Try this beautiful problem from Integer based on Prime number useful for ISI BStat Entrance.
Prime number | ISI BStat Entrance | Problem no. 70
The number of integers \(n>1\), such that n, n+2, n+4 are all prime numbers is ......
Zero
One
Infinite
More than one,but finite
Key Concepts
Number theory
Algebra
Prime
Check the Answer
Answer: One
TOMATO, Problem 70
Challenges and Thrills in Pre College Mathematics
Try with Hints
taking n=3, 5, 7, 11, 13, 17....prime numbers we will get
Case of n=3
n= 3
n+2=5
n+4=7
Case of n=5
then \(n\)=5
n+2=7
n+4=9 which is not prime....
Case of n=7,
then n=7
n+2=9 which is not prime ...
n+4=11
Can you now finish the problem ..........
We observe that when n=3 then n,n+2,n+4 gives the prime numbers.....other cases all are not prime.Therefore any no can be expressed in anyone of the form 3k, 3k+1 and 3k+2.
can you finish the problem........
If n is divisible by 3 , we are done. If the remainder after the division by 3 is 1, the number n+2 is divisible by 3. If the remainder is 2, the number n+4 is divisible by 3
The three numbers must be primes! The only case n=3 and gives\((3,5,7)\)
Prime number Problem | ISI BStat | TOMATO Objective 96
Try this beautiful problem from Integer based on Prime number useful for ISI B.Stat Entrance.
Prime number | ISI B.Stat Entrance | Problem no. 96
The number of different prime factors of 3003 is.....
2
15
7
16
Key Concepts
Number theory
Algebra
Prime numbers
Check the Answer
Answer: 16
TOMATO, Problem 96
Challenges and Thrills in Pre College Mathematics
Try with Hints
At first, we have to find out the prime factors. Now \(3003\)=\(3 \times 7 \times 11 \times 13\). but now it can be expressed as another prime number also such as \(3003=3 \times 1001\). So we have to find different prime factors.
Can you now finish the problem ..........
Now, if you have a number and its prime factorisation, \(n={p_1}^{m_1} {p_2}^{m_2}⋯{p_r}^{m_r}\) you can make divisors of the number by taking up to \(m_1\) lots of \(p_1\), up to \(m_2\) lots of \(p_2\) and so on. The number of ways of doing this is going to be\( (m_1+1)(m_2+1)⋯(m_r+1)\).
can you finish the problem........
for the given case \(3003\) has \(2^4=16 \)divisors.
Remainder Problem | ISI-B.Stat Entrance | TOMATO 90
Try this beautiful problem from Integer based on Remainder useful for ISI B.Stat Entrance.
Remainder Problem | ISI B.Stat Entrance | Problem-90
The remainder when \( 3^{12} +5^{12}\) is divided by 13 is......
2
1
4
3
Key Concepts
Division algorithm
Divisor
Number theory
Check the Answer
Answer: 2
TOMATO, Problem 90
Challenges and Thrills in Pre College Mathematics
Try with Hints
The given number is \( 3^{12} +5^{12}\)
we have to check if it is divided by 13 what will be the remainder? if we express the number in division algorithm form then we have........\( 3^{12} +5^{12}=((3)^3)^4+((5)^2)^6)=(27)^4 +(25)^6\)=\(((13 \times 2+1)^4+(13 \times 2-1)^6)\)
Can you now finish the problem ..........
Remainder :
Clearly if we divide \(((13 \times 2+1)^4+(13 \times 2-1)^6)\) by 13 then from \((13 \times 2+1)^4\) , the remainder be 1 and from \((13 \times 2-1)^6)\), the remainder is 1
Number counting | TOMATO ISI BStat Objective Problem 56
Try this beautiful problem Based on Number counting useful for ISI B.Stat Entrance.
Number counting| ISI B.Stat Entrance |Problem 56
In a group of 120 persons there are 70 Bengalis,35 Gujaratis and 15 Maharashtrians.Further 75 persons in the group are Muslims and the remaining are Hindus.Then the number of Bengali Muslims in the group is
between 10 and 14
between 15 and 19
exactly 20
25 or more
Key Concepts
Number counting
Algebra
Set
Check the Answer
Answer: 25 or more
TOMATO, Problem 56
Challenges and Thrills in Pre College Mathematics
Try with Hints
Find the numbers of Hindus
Can you now finish the problem ..........
Assume that all hindus are in bangalis to find the minimum numbers of muslims persons
can you finish the problem........
Given that total number of person =120
75 persons are muslims
Therefore number of Hindus are(120-75)=45
There are 70 Bengalis and we assume that 45 hindus are in Bengalis to find the minimum number of muslims.
Therefore Bengali Muslims =(70-45)=25
Hence the number of Bengali Muslims in the group is 25 or more
