Number Sense Section 3 Formulas/Tricks

Most formulas/tricks needed from Section 3 of Bryant Heath's Number Sense guide: https://bryantheath.com/files/2018/04/Heath_NSTricks_revA.pdf

How to find the GCD of two numbers?

Find the remainder of dividing the larger number by the smaller, then find the remainder of dividing the smaller number by the previous remainder, etc.

How to find the LCM of two numbers?

Divide the numbers multiplied together by the GCD

How to find the GCD of three numbers?

Calculate the GCD of the LCM of two numbers and the third number

Sum of Coefficients of (ax+by)^n

(a+b)^n

Zeroth row of Pascal’s triangle

1

First row of Pascal’s triangle

1 1

Second row of Pascal’s triangle

1 2 1

Third row of Pascal’s triangle

1 3 3 1

Fourth row of Pascal’s triangle

1 4 6 4 1

Fifth row of Pascal’s triangle

1 5 10 10 5 1

Sixth row of Pascal’s triangle

1 6 15 20 15 6 1

Units digit rotation for powers of 2

2 4 8 6

Units digit rotation for powers of 3

3 9 7 1

Units digit rotation for powers of 4

4 6

Units digit rotation for powers of 5

5

Units digit rotation for powers of 6

6

Units digit rotation for powers of 7

7 9 3 1

Units digit rotation for powers of 8

8 4 2 6

Units digit rotation for powers of 9

9 1

log(2)

.3

log(5)

.7

ln(2)

.7

ln(10)

2.3

Complex Conjugate

a-bi

Complex Modulus

sqrt(a^2+b^2)

Complex Argument

arctan(b/a)

(a+bi)·(c+di)

(ac-bd)+(ad+bc)i

(a+bi)^(−1)

(a-bi)/(a^2+b^2)

Inverse Function of (ax+b)/(cx+d)

(-dx+b)/(cx-a) (flip and negate a and d)

A n B

Intersection (number of terms in common)

A U B

Union (number of terms all together)

A’

Complement of A (all elements not in A that are in some other set)

Number of subsets for a set with n elements

2^n

Number of proper subsets for a set with n elements

2^n-1

Number of elements in a set’s power set

Number of subsets

0.aaaaa…=

a/9

0.ababab…=

ab/99

0.abbbbb…=

(ab-a)/90

0.abcbcbc…=

(abc-a)/990

Which part of the exponent a^b can you not use modular arithmetic to simplify?

b

1×1!+2×2!+…+n*n! =

(n+1)!-1

Wilson’s Theorem

For prime p, (p-1)! ~=(p-1) mod p

Special Integral that always equals 0

Integral from -a to a of an odd function