The given question is an easy to moderate level difficulty question combining concepts in Arithmetic Progression and elementary number theory - LCM in particular.
Question 23: In the first 1000 natural numbers, how many integers exist such that they leave a remainder 4 when divided by 7, and a remainder 9 when divided by 11?
The first sequence: Numbers leaving a remainder of 4 when divided by 7: 4, 11, 18, 25, 32, 39, 46, 53, 60, 67, ....
The second sequence: Numbers leaving a remainder of 9 when divided by 11: 9, 20, 31, 42, 53, 64, .....
From the listing of the two sequences we can identify the first number that is a part of the both the sequences is 53.
The common difference of the first sequence is 7 and that of the second sequence is 11.
Elements common to both the sequences will have a common difference that is the LCM of 7 and 11.
77 is the LCM of 7 and 11.
Every 77th number after 53 will be a term common to both the series.
So, the terms that are common to both the arithmetic sequences can be expressed as 77k + 53.
Because we are interested in the first 1000 natural numbers, k will take values from 0 to 12.
i.e., a total of 13 values.
Register in 2 easy steps and
Start learning in 5 minutes!
1. Number Systems | Types of Numbers | Chart
2. Number Properties | Rational & Irrational Numbers
3. GMAT Number Properties | Indices & Rule of Exponents
4. Number Systems | Surds & Conjugates
5. Number Properties | Tests of divisibility
6. Number Properties | How to check whether a number is prime?
7. Number Properties | How to prime factorize a number?
8. GMAT Number Theory | Prime factorization | Properties of squares & cubes
9. Number Properties | What is HCF? | How to find HCF?
10. Number Properties | What is LCM? | How to find LCM?
11. 3 important properties of LCM & HCF | LCM & HCF of fractions
12. Number Properties | When to use LCM and HCF?
13. Number Theory | How to find number of factors?
14. Number Theory | Number of ways to express as a product of 2 factors
15. Number Theory | Sum of all factors of a number
16. Number Theory | Product of all factors of a number
17. Number Theory | Remainders of sum & product
18. Number Theory | Remainder of dividing xn by 'd'
19. Polynomials | Remainder when a monomial divides it
20. Number Theory | Highest power of a prime that divides factorial of 'n'
21. Number Theory | Highest power of a composite number that divides factorial of 'n'
22. Number Theory | Number of trailing zeroes in a number
23. Number Theory | Unit digit of higher powers of numbers
Copyrights © 2016 - 21 All Rights Reserved by Wizako.com - An Ascent Education Initiative.
Privacy Policy | Terms & Conditions
GMAT® is a registered trademark of the Graduate Management Admission Council (GMAC). This website is not endorsed or approved by GMAC.
GRE® is a registered trademarks of Educational Testing Service (ETS). This website is not endorsed or approved by ETS.
SAT® is a registered trademark of the College Board, which was not involved in the production of, and does not endorse this product.
Wizako - GMAT, GRE, SAT Prep
An Ascent Education Initiative
48/1 Ramagiri Nagar
Velachery Taramani Link Road.,
Velachery, Chennai 600 042. India
Phone: (91) 44 4500 8484
Mobile: (91) 95000 48484
WhatsApp: WhatsApp Now
Email: learn@wizako.com
Leave A Message