Question

​Suyín has 135 pennies and 225 nickels. She wants to place them all in stacks so that each stack has the same number of​ coins, and each stack contains only one denomination of coin. What is the greatest number of coins that she can place in each​ stack?

Answer 1

$\large\underline{\sf{Solution-}}$

➢ Given that,

Suyin had 135 pennies and 225 nickels.

She wants to place them all in stacks so that each stack has the same number of coins, and each stack contains only one denomination of coin.

➢ It means, we have to find the largest number which divides 135 and 225 exactly.

And, we know that

➢ That number is HCF ( 135, 225 ).

CONSIDER,

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 135}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{3}}}&{\underline{\sf{\:\:135 \:\:}}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:45 \:\:}} \\\underline{\sf{3}}&\underline{\sf{\:\:15\:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5 \:\:}} \\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

So,

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 135 = {3}^{3} \times 5}$

Now,

CONSIDER

$\purple{\rm :\longmapsto\:Prime \: Factorization \: of \: 225}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{3}}}&{\underline{\sf{\:\:225 \:\:}}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:75 \:\:}} \\\underline{\sf{5}}&\underline{\sf{\:\:25\:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5 \:\:}} \\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

So,

$\purple{\rm :\longmapsto\:Prime \: Factorization \: of \: 225 = {3}^{2} \times {5}^{2} }$

Thus, we have

$\purple{\rm :\longmapsto\:Prime \: Factorization \: of \: 225 = {3}^{2} \times {5}^{2} }$

$\red{\rm :\longmapsto\:Prime \: Factorization \: of \: 135 = {3}^{3} \times 5}$

So, it implies

$\green{\bf :\longmapsto\: HCF(135 , \: 225) = {3}^{2} \times 5 = 45}$

Hence,

Suyan, can make stacks of 45 coins of each type so that each stack has the same number of coins, and each stack contains only one denomination of coin.

Additional Information :-

Practice problem of same type. Try yourself !!

Question :- 1 In a seminar, the number of participants in Hindi, English ane Mathematics are 60, 84 and 108 respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject.

Answer :- 10

Question :- 2 Three sets of English, Hindi and Mathematics books have to be stacked in such a way that each stack has the same number of books and each stack contains topic wise book and height of each stack is same. The number of English books, Hindi books and Mathematics books is 96, 240 and 336 respectively. Find the greatest number of books that can be placed in each stack assuming that thickness of each book is same.

Answer :- 48