Chapter 3 Finding Common Ground (Class 7 - Latest Maths NCERT (Ganita Prakash II) Solutions)

Question. List all the factors of the following numbers:

(a) $90$

(b) $105$

(c) $132$

(d) $360$ (this number has $24$ factors)

(e) $840$ (this number has $32$ factors)

Question. Find the common factors and the HCF of the following numbers:

(a) $50, 60$

(b) $140, 275$

(c) $77, 725$

(d) $370, 592$

(e) $81, 243$

How do we directly find the HCF without listing all the factors?

Question 1. Find the HCF of the following numbers:

(a) $24, 180$

(b) $42, 75, 24$

(c) $240, 378$

(d) $400, 2500$

(e) $300, 800$

Question 2. Consider the numbers $72$ and $144$.

Suppose they are factorised into composite numbers as: $72 = 6 \times 12$ and $144 = 8 \times 18$. Seeing this, can one say that these two numbers have no common factor other than $1$? Why not?

Question. Find the LCM of the following numbers:

(a) $30, 72$

(b) $36, 54$

(c) $105, 195, 65$

(d) $222, 370$

Question 1. Make a general statement about the HCF for the following pairs of numbers. You could consider examples before coming up with general statements. Look for possible explanations of why they hold.

(a) Two consecutive even numbers

(b) Two consecutive odd numbers

(c) Two even numbers

(d) Two consecutive numbers

(e) Two co-prime numbers

Share your observations with the class.

Question 2. The LCM of $3$ and $24$ is $24$ (it is one of the two given numbers).

(a) Find more such number pairs where the LCM is one of the two numbers.

(b) Make a general statement about such numbers. Describe such number pairs using algebra.

Question 3. Make a general statement about the LCM for the following pairs of numbers. You could consider examples before coming up with these general statements. Look for possible explanations of why they hold.

(a) Two multiples of $3$

(b) Two consecutive even numbers

(c) Two consecutive numbers

(d) Two co-prime numbers

Question 1. In the two rows below, colours repeat as shown. When will the blue stars meet next?

Question 2. (a) Is $5 \times 7 \times 11 \times 11$ a multiple of $5 \times 7 \times 7 \times 11 \times 2$?

(b) Is $5 \times 7 \times 11 \times 11$ a factor of $5 \times 7 \times 7 \times 11 \times 2$?

Question 3. Find the HCF and LCM of the following (state your answers in the form of prime factorisations):

(a) $3 \times 3 \times 5 \times 7 \times 7$ and $12 \times 7 \times 11$

(b) $45$ and $36$

Question 4. Find two numbers whose HCF is $1$ and LCM is $66$.

Question 5. A cowherd took all his cows to graze in the fields. The cows came to a crossing with $3$ gates. An equal number of cows passed through each gate. Later at another crossing with $5$ gates again an equal number of cows passed through each gate. The same happened at the third crossing with $7$ gates. If the cowherd had less than $200$ cows, how many cows did he have? (Based on the folklore mathematics from Karnataka.)

Question 6. The length, width, and height of a box are $12\text{ cm}$, $18\text{ cm}$, and $36\text{ cm}$ respectively. Which of the following sized cubes can be packed in this box without leaving gaps?

(a) $9\text{ cm}$

(b) $6\text{ cm}$

(c) $4\text{ cm}$

(d) $3\text{ cm}$

(e) $2\text{ cm}$

Question 7. Among the numbers below, which is the largest number that perfectly divides both $306$ and $36$?

(a) $36$

(b) $612$

(c) $18$

(d) $3$

(e) $2$

(f) $360$

Question 8. Find the smallest number that is divisible by $3, 4, 5$ and $7$, but leaves a remainder of $10$ when divided by $11$.

Question 9. Children are playing ‘Fire in the Mountain’. When the number $6$ was called out, no one got out. When the number $9$ was called out, no one got out. But when the number $10$ was called out, some people got out. How many children could have been playing initially?

(a) $72$

(b) $90$

(c) $45$

(d) $3$

(e) $36$

(f) None of these

Question 10. Tick the correct statement(s). The LCM of two different prime numbers $(m, n)$ can be:

(a) Less than both numbers

(b) In between the two numbers

(c) Greater than both numbers

(d) Less than $m \times n$

(e) Greater than $m \times n$

Question 11. A dog is chasing a rabbit that has a head start of $150$ feet. It jumps $9$ feet every time the rabbit jumps $7$ feet. In how many leaps does the dog catch up with the rabbit?

Question 12. What is the smallest number that is a multiple of $1, 2, 3, 4, 5, 6, 8, 9, 10$? Do you remember the answer from Grade $6$, Chapter $5$?

Question 13. Here is a problem posed by the ancient Indian Mathematician Mahaviracharya ($850$ C.E.). Add together $\frac{8}{15}$, $\frac{1}{20}$, $\frac{7}{36}$, $\frac{11}{63}$ and $\frac{1}{21}$.

What do you get? How can we find this sum efficiently?