## 7th and 8th Grade, October, 2019

C.1257.

Using the first six positive digits, we can write the following equality using each digit exactly once so that on both sides of the equality you have only exponential expressions. (There has to be an exponent written in each expression, and you are allowed to use only one-digit numbers.)

Find as many similar equalities using the first eight positive digits as you can.

C.1258. There are 700 hymns in the book of Hymns, numbered from 1 to 700. Every Sunday at the service the congregation sings 4 different hymns from this book. The numbers of the hymns are posted on a board where they use a small card for each digit to put the numbers together. They can use the card with the digit 6 on it also as a digit 9. At least how many cards should they make for each digit so that they could post any set of 4 hymn numbers on the board for the congregation?

C.1259. There are 10 coins that look alike, but 2 of them are fake. The fake ones are lighter than the real ones, but both have the same weight. You are given a balance scale, and you should determine which coins are fake. At least how many weighings do you need to find the fake coins?

C.1260. Find the smallest positive composite number that can be written as:

a)  the product

b) the sum

of two different prime numbers in two different ways.

C.1261. How many 6-digit numbers of the form ABAABA can be written as the product of 4 prime numbers?

C.1262. Laslie’s calculator does not work properly. He realized that two number keys’ functions got switched: if he presses one of these two numbers, the calculator reacts as if he pressed on the other number, and vice versa. If he punches in 8×45, the result is 360. If he punches in 6×87 the result is 602. Which two number keys got switched?

C.1263. We made a big cube out of small, unit cubes. We painted every side of the big cube blue, and then we took the big cube apart. There were then 72 small cubes each with exactly 2 sides painted blue. How many small cubes have no more than one blue side?

C.1264. The first four digits of a 6-digit number are 9, 7, 5, 3, in this order. What are its last two digits if the number is divisible by 7 and 13?