practice problems pending - easy

Q1). A number N in base 10, is 503 in base b and 305 in base b+2. What is the product of the digits of N?

S1.
N = 248
Answer = 64

Q2).  Let (ABCD) be a rectangle, in which (AB + BC + CD = 20) and (AE = 9), where (E) is the midpoint of the side (BC). Find the area of the rectangle.
S2.
2x + y = 20______[1]
81 = x^2 + y^2/4______[2]
We just need to find xy.
Square the first equation and you will 4x^2 + y^2, replace it using 2 and you will get x.y = 19.

Q3). What is the least positive integer by which 2^5. 3^6. 4^3. 5^3.6^7 should be multiplied so that the product is a perfect square?

S3.
Write it as 2^(5+6+7).3^(6+7).5^3 = 2^18.3^13.5^3 So the answer is 15.

Q4). Find the number of integer solutions to

||x| - 2020| < 5
S4.

-5 < |x| - 2020 < 5
2015 < |x| < 2025
=> x has 18 possible integer solutions when |x| = 2016 to 2024

Q5). The product (55 * 60 * 65) is written as the product of five distinct positive integers. What is the least possible value of the largest of these integers?

S5.
Do the prime factorization:
2^2 * 3^1 * 5^3 * 11^1 * 13^1

So the largest integer can't be less than 13.

Let's try with 13 as the largest.
Remaining:
2^2 * 3^1 * 5^3 * 11^1

= 11 * 6 * 10 * 25
So it doesn't work.

Let's try with 15:
Remaining:
2^2 *  5^2 * 11^1 * 13^1

13 * 11 * 10 * 10
But all have to be distinct, and 10 repeats here.

Let's try 20:
Remaining:
3^1 * 5^2 * 11^1 * 13^1

= 11 * 13 * 15 * 5

So it works.
Answer = 20



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