78.6k views
3 votes
Find the smallest positive integer (N) with the following property: of the three numbers (N), (N+1), and (N+2), one of them is divisible by (2^2), one of them is divisible by (3^2), one is divisible by (5^2), and one is divisible by (7^2).

a) 630
b) 882
c) 1260
d) 1764

User Bubuxu
by
8.0k points

1 Answer

0 votes

Final answer:

The smallest positive integer N that satisfies the given divisibility conditions by the squares of the primes 2, 3, 5, and 7 is 1764. It is the only option where N, N+1, and N+2 are each divisible by a different one of these squares.

Step-by-step explanation:

The question requires finding the smallest positive integer N such that N, N+1, and N+2 are each divisible by the squares of different primes: 2, 3, 5, and 7. To solve this problem, we need to consider the divisibility conditions for each number:

  • A number is divisible by 2^2 (4) if its last two digits form a number divisible by 4.
  • A number is divisible by 3^2 (9) if the sum of its digits is divisible by 9.
  • A number is divisible by 5^2 (25) if its last two digits are 00, 25, 50, or 75.
  • A number is divisible by 7^2 (49) only if it satisfies the divisibility rule of 49, which is more complex and typically requires direct division to verify.

Through testing each option, we determine that:

  • 630 is divisible by 5^2 and 7^2, but 631 and 632 are not divisible by 2^2 or 3^2.
  • 882 is divisible by 2^2, 883 is not divisible by 3^2 or 5^2, and 884 is divisible by 2^2 again, failing the requirement.
  • 1260 is divisible by 2^2, 1261 is not divisible by 3^2, and 1262 is divisible by 2^2, so it also fails.
  • 1764 is divisible by 7^2, 1765 is divisible by 5^2, and 1766 is divisible by 2^2, meeting all the requirements.

Therefore, the smallest positive integer N with the given property is 1764.

User Jtooker
by
7.8k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories