Question

Find the largest number that divides 1086 and 1125 leaving remainder of 6 and 5 respectively

Answer 1

Answer :-

40

Solution :-

If 1086 and 1125 leaves remainder 6 and 5 respectively when divided by largest number.

Now, to know that largest number we must know the numbers which are divisible by largest no.

1086 leaves remainder 6

So, the number divisible by largest no. = 1086 - 6 = 1080

1125 leaves remainder 5

So, the number divisible by largest no. = 1125 - 5 = 1120

Now, the largest no. will be the HCF of 1120 and 1080

Finding HCF of 1120 and 1080

Method I : Prime factorisation method

Prime factorisation of 1080

$\begin{array}{r | l} 2 & 1080 \\ \cline{1-2} 2 & 540 \\ \cline{1-2} 2 & 270 \\ \cline{1-2} 3 & 135 \\ \cline{1-2} 3 & 45 \\ \cline{1-2} 3 & 15 \\ \cline{1-2} 5 & 5 \\ \cline{1-2} & 1 \end{array}$

Prime factorisation of 1120

$\begin{array}{r | l} 2 & 1120 \\ \cline{1-2} 2 & 560 \\ \cline{1-2} 2 & 280 \\ \cline{1-2} 2 & 140 \\ \cline{1-2} 2 & 70 \\ \cline{1-2} 5 & 35 \\ \cline{1-2} 7 & 7 \\ \cline{1-2} & 1 \end{array}$

1080 = 2³ * 5 * 3³

1120 = 2^5 * 5 * 7

HCF of 1080 and 1120 = Product of common prime factors with smallest power = 2³ * 5 = 8 * 5 = 40

Method II : Euclid's algorithm

Euclid's algorithm - a = bq + r [ Where r is between 0 and b and r is also equal to 0 ]

1120 = (1080 * 1) + 40

1080= (40 * 27) + 0

As, r = 0 divisor at this stage is HCF

So, HCF = 40

Therefore 40 is the largest number that divides 1086 and 1125 leaves remainder 6 and 5 respectively.