Question

The number of prime numbers between 11 and 60 is ____.
The numbers 29 and ____ are twin primes.
3753 is divisible by 9 and hence divisible by ____.
The number of distinct prime factors of the smallest 4-digit number is ____.
The sum of distinct prime factors of 30 is ____.

Answer 1

Hint: First list all the prime numbers which lie between 11 and 60. Then count the numbers.
The twin prime of 29 is 31.
Factorize the smallest 4-digit number and count the distinct prime factor.
Factorize the number 30 and add all the distinct prime factors.

Complete step-by-step answer:
The prime numbers between 11 and 60 are 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 and 59.
There are a total of 12 prime numbers that lie between 11 and 60.
Hence, the number of prime numbers between 11 and 60 is 12.
A twin prime is a prime number that differs by 2 from another prime number.
The numbers that differ by 2 from 29 are 27 and 31.
As 27 is not a prime number only 31 is a prime number.
Hence, the numbers 29 and 31 are twin primes.
As 3753 is divisible by 9 and 9 is divisible by 3.
Hence, 3753 is divisible by 9 and hence divisible by 3.
The smallest 4-digit number is 1000.
The smallest number by which 1000 is divisible is 2. Then,
$1000 = 2 \times 500$
The smallest number by which 500 is divisible is 2. Then,
$1000 = 2 \times 2 \times 250$
The smallest number by which 250 is divisible is 2. Then,
$1000 = 2 \times 2 \times 2 \times 125$
The smallest number by which 125 is divisible is 5. Then,
$1000 = 2 \times 2 \times 2 \times 5 \times 25$
The smallest number by which 25 is divisible is 5. Then,
$1000 = 2 \times 2 \times 2 \times 5 \times 5 \times 5$
So, the distinct prime factors of 1000 are 2 and 5.
Hence, the number of distinct prime factors of the smallest 4-digit number is 2.
The smallest number by which 30 is divisible is 2. Then,
$30 = 2 \times 15$
The smallest number by which 15 is divisible is 3. Then,
$30 = 2 \times 3 \times 5$
So, the distinct prime factors of 1000 are 2 and 5.
The sum of distinct prime factors is,
$2 + 3 + 5 = 10$
Hence, the sum of distinct prime factors of 30 is 10.

Note: Students must exclude 11 as it is mentioned the prime numbers between 11 and 60.
The students must know the meaning of twin primes to find the twin primes of 29.
The student must not go for the division of 3753 by 9 to find other factors of 3753.
Students must not count all prime factors or all factors of 4-digit numbers as it is mentioned about distinct prime factors.