Answer

Verified

432.6k+ views

Hint: To find the smallest number which when divided by \[35,56,91\] leaves remainder \[7\] in each case, calculate the LCM of \[35,56,91\] and add \[7\] to the LCM of \[35,56,91\].

Complete step-by-step answer:

We have to find the smallest number which when divided by \[35,56,91\] leaves remainder \[7\] in each case. To do so, we will firstly evaluate the smallest number which is a multiple of each of \[35,56,91\].

Thus, we will evaluate the LCM of \[35,56,91\].

To find the LCM of three numbers, we will use the prime factorization method. We will write the prime factorization of each of the numbers in exponential form. Then, we will align the common prime factor base whenever possible. For the numbers with a common prime factor base, select the prime number that has the highest power. The prime factor with highest power implies that it occurs the most in the list. If a distinct prime factor has no matching prime factor base in the list, include this factor with its exponent in the collection of numbers. Multiply all the numbers which were collected earlier to get the LCM of three numbers.

Factorizing each of the numbers, we have \[35=5\times 7,56={{2}^{3}}\times 7,91=7\times 13\] .

Thus, the LCM of \[35,56,91\] is \[7\times 5\times {{2}^{3}}\times 13=3640\].

The LCM of \[35,56,91\] is \[3640\].

We have to find the smallest number which leaves a remainder \[7\] when divided by \[35,56,91\], add \[7\] to the LCM of numbers \[35,56,91\].

Thus, we have \[3640+7=3647\].

Hence, the smallest number which leaves a remainder \[7\] when divided by \[35,56,91\] is \[3647\].

Note: Least Common Multiple (LCM) of any two numbers is the smallest positive number that is divisible by both the numbers. We can also find LCM of two numbers using list method. We should carefully calculate the LCM of three numbers; otherwise we will get an incorrect answer.

Complete step-by-step answer:

We have to find the smallest number which when divided by \[35,56,91\] leaves remainder \[7\] in each case. To do so, we will firstly evaluate the smallest number which is a multiple of each of \[35,56,91\].

Thus, we will evaluate the LCM of \[35,56,91\].

To find the LCM of three numbers, we will use the prime factorization method. We will write the prime factorization of each of the numbers in exponential form. Then, we will align the common prime factor base whenever possible. For the numbers with a common prime factor base, select the prime number that has the highest power. The prime factor with highest power implies that it occurs the most in the list. If a distinct prime factor has no matching prime factor base in the list, include this factor with its exponent in the collection of numbers. Multiply all the numbers which were collected earlier to get the LCM of three numbers.

Factorizing each of the numbers, we have \[35=5\times 7,56={{2}^{3}}\times 7,91=7\times 13\] .

Thus, the LCM of \[35,56,91\] is \[7\times 5\times {{2}^{3}}\times 13=3640\].

The LCM of \[35,56,91\] is \[3640\].

We have to find the smallest number which leaves a remainder \[7\] when divided by \[35,56,91\], add \[7\] to the LCM of numbers \[35,56,91\].

Thus, we have \[3640+7=3647\].

Hence, the smallest number which leaves a remainder \[7\] when divided by \[35,56,91\] is \[3647\].

Note: Least Common Multiple (LCM) of any two numbers is the smallest positive number that is divisible by both the numbers. We can also find LCM of two numbers using list method. We should carefully calculate the LCM of three numbers; otherwise we will get an incorrect answer.

Recently Updated Pages

The branch of science which deals with nature and natural class 10 physics CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE

What organs are located on the left side of your body class 11 biology CBSE