Find the largest number that divides 2053 and 967 and leaves a remainder of 5 and 7 respectively.

Foundation

Mathematics Foundation

HCF and LCM

Properties of Real Number

Remainder Theorem

Question

Find the largest number that divides

2053

and

967

and leaves a remainder of

5

and

7

respectively.

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

Given that on dividing

2053

, there is a remainder of

5

.
This means that

2053 - 5 = 2048

is exactly divisible by the required number.
Similarly,

967 - 7 = 960

is exactly divisible by the required number.

2048 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2

= 2^{11}

and

960 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5

= 2^{6} \times 3 \times 5

The smallest common power in both prime factorization

= 2^{6}

= 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64

So, H.C.F

= 64